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er
r
er
o
e
o
e rre
,
,
’
o r an
o en
,
o
n
re
c o rre c
n
n
an
a
o
av e
e
e ve r
o
ac
ar e
a
e
ar
an
e co n
no
on
o
e
or
an
re a
e are
or
e
va
n
en
ea
ue
an
n
e
e
o
ne
n
s
a
e
.
o u r co
n
ar
c
e
e
o
on
ue
e ec
an
an
n
,
e
an
e, o u r
e
o
our
o
or
o e
ar
M a y , 1 88 7
on
o
e
a
or o
ro e
,
re
ex
ee
a
.
.
ra e u
e
so
c
n
ro
o
e
o
or
e
er
n
o
e
.
.
o
a
er
,
n e er n
n
on
a
e,
o
n
ec
ro o
.
n
ne
n
or
e
.
re v
n
o
ev
e
a
a er c
r en
an
t en
e co
a n
r
o
e
e
co n
an
,
.
s
our
o
a n
o
or
s
ro
ro
er
r
n
on
o c cas o n a
,
.
e
a
ue
are
o
ev
e
e
a
ave
e
e ra ,
a
an
re n
a
an
e
e
on
.
S H A LL
H
.
S R
.
PR E FA C E TO TH E T H I R D
.
,
KN I GH T
.
E D I T ION
.
is diti th t t d mpl s
s u b s t ti l]
th
s m i p e i u s di ti s b t f w ti l s
mpl s h b
s t d ll th
ifi d g m
b
ll ti
ls
dd d
f th
h u d d Mi
l
W h
h
l
s E m
l
h
w
l
l
s
i
i
u
s
d
l
w
b
f
d
f
f
d
p
mpl s h b
s tu d ts Th s
s l t d mi ly b t
lu si ly f mS h l s h i p
t
S
t H us p p s
h
mu h
b
t
i llu s t t
y p t f th
t k
s u bj t d t f i ly p s t th p i ip l U i si ty d
i E mi ti s
C i il S
I N th
e
e
as
e
a
e en
re c a
e
ave
,
a
en
v
a
ve
,
o
e rv c e
,
1 88 9
a r
.
ave
o ar
a
en
re
na
re e n
on
o
.
n
o
u
ar
e u
en a e
r nc
a
e
ar
n
ve r
a
r
sc e
van c e
a n
e ve r
a
e
or a
o
a
c e
re
n
e ec e
ra e
e
an
e e n v er
ee n
or
e
re e
O
e
c
a
ave
on
are
e
u
,
e
ec
e
e en
xa
exa
co
exa
as
on
c
ro
an
M a rc h
e
a
e xa
an
ex
e
e
e e
c are
ec
a
e
.
ex c
c
an
o
e
r v o
n
xa
an e o u
n o
on
u
a
er
o
e
an
C O N TE N T S
C HAPT E R I
Co
mm
en s
R atio
u
d in co
rab le an
mm
en s
u
f greater an d les s in eq
o
a g
e
b
f
d
n
a
+
qc
n
+
u
rab le
q
ality
n
u
RAT I O
.
an
.
.
titie s
+
7e
b1 + b2 + b3
multipli ti
E lim
i
t f th
E
mpl I
C ro
ss
on
ca
es
xa
lin ear eq
ree
O
n an
u
atio n s
.
C H A PT E R
D
d Pro p o s itio n s
b e t een algeb rai cal
efi n itio n s an
mp
Co
aris o n
C as e
o
f in co
mpl
E xa
es
w
mm
en s
II
u
rab le
q
u
an
an
II
d geo
In
v
v
ers e
Jo in t
If A
u
I ll
v
c:
fi
h
B, t
(I
e r c al
efi n
B
s
en
wh
en
0
es
.
itio n s
.
III
V
.
A RI AT I O N
B
m
A=
ariatio n
mpl
d
ariatio n
s tratio n s
E xa
mt i
titie s
C HAPT E R
H A
PR OPO RT I O N
.
c o n s tan
mp l
E xa
.
II I
0 is
.
es o n
j
t,
an
o in
t
d A
v
at:
0
a ria t io n
wh
en
B
is
.
CO
C H APT E R
IV
N TE NT S
.
A RI T H ME T I C AL P R O G RE SS I O N
.
.
ith mti
mf t m f
ul
Fu d m t l f m
f ith mti m
I
ti
E
mpl I V
Su
o
n
n
er
s o
an ar
a
or
a
e
en
a
n s er
on O
xa
es
D u
isc
mpl
E xa
IV b
es
Su
mf
C H APT E R
f geo
o
an
mf
mpl V
m
V
G E O ME T RI C AL P R O G RE SS I O N
.
u
a
.
.
m
ter
n
2s
n
0
re
du
c tio n o
arith
f an
s o
.
s erie s
s erie s
e r ca
eo
e
n
n
es
o
d)
e r ca
eo
a
s o
er
n
mf
(2a
e an s
e r c
Pro o f o f r le fo r th e
Su
f an
mt i m
t m f g mt i l
i fi it g mt i l
o
E xa
o
?
.
.
s ertio n o
Su
f ro o ts
o
i/
In
a.
.
s s io n
e an s
c
e
ar
c al s erie s
e
fa
mti
e
co
-
re c
ge o
u
rrin
mt i
g
d
ml
ec i
a
e r c s erie s
E xa p les V b
.
C H APT E R V I
HA R M O NI C AL P R O GRE SS I O N T H E O REM S C O NN E C T E D
W I T H TH E P R O G RE SS I O N S
.
.
.
R ecip ro c als
H ar o n ic
o
fq
m m
mul
Fo r
Su
2
an
titie s in H
mf
mf
u
l tio n
so
u
u
O
sq
o
c
are s o
b es
o
.
f th e
f th e
.
u
n at
u
n at
.
u
es
a
.
s
o
n
ral n
ra
on a
ra
o n a rec an
e e
xa
.
.
r an
ar
.
um
um
s
b ers
b ers
o n a sq
u
are
b as e
as e
ar
as e
ra
es
.
.
C HAPTE R V II
E x p lan atio n
E xa
in A P
.
.
ra
n co
are
.
ral n
mpl VI
b
N um
f h t i py m
id
Py m
id
t i gul
b
Py m
id
t
b
gu l
I
mpl t py mid
E
mpl VI b
er o
.
gA M , C M , H M
o f q es tio n s in Pro gres s io n
n o tatio n
E xa
P
.
e an
a
e c o n n ec tin
H in t s fo r
Su
u
o
mp l V II
es
f s y s te
.
a
m
s o
E x p re s s io n
o
f an in tegral
E x p re s s io n
o
fa
d
OF N OT AT I O N
p ro
d
f n o tatio n
.
ra
CAL E S
S
.
n
um
b er in
ix frac tio n in
a
a
p ro
po se
po se
d
s c ale
s c ale
.
C O NT E NT S
d
by
l
r—
o
u
dv
“
ca s tin
f i is ib ility b y
b
ple s
m V II
E xa
C be
mg th
m
t
ro o
t
d
e
en o
es
a
n ar
If
du u
S u
Pw
ub
P w
l
f p ro
s o
o
e ro o
t
S
.
en Cl
h
du
r
-
0, b
en a
is
ct
es
o
ro o
mti
Fo r
d
a
b
.
on
ts
ud
o
e an
mpl
E xa
es
Fo r real
f eq
h
u
eq
al
D
/
b
/
N
to p ro
2
; 1 + w+ w
IX
N C
l
''
IX
v u
al
e
a
.
du
h
h
C o n ditio n t
ct o
f
mduli
o
T H E O RY OF Q U AD R A T I C
hv m h w
th e
ro o
a e
n ary ro o
a
c t o f ro o
en
f
ts
o
in
Sign
th e
a
q
-
ud
an
C o n itio n t
t
mpl
es
ts
t
o ro o
v
h ud
gi
are
a ratic s
en
l
o
b e ( 1)
eq
re c ip ro c als
u
al
mg
in
a n
f r th e
e x p re s s io n a x
2
+ b x + c h as in gen
eral
th e
i
m
sa
e
e x ce p tio n s
;
b
.
at acc
at a
.
ts
.
ffn n c tio n
IX
E Q U AT I O N S
ts
;
ts z
ro o
( 2)
,
t
o re
2
,
va ria b le , ra ti o n a l
2
2
+ hx y + b y
i n tegra l fu n c ti o n
29 50 + 2fy +
c
my b
a
lin ear fac to rs
E xa
.
0
:
mgi
at io n s
es o
.
efi n i tio n s o
ro o
ANT I T I E S
fl
TH E
.
u
Opp o s ite
d
mpl s IX
d
QU
.
du
g
u
wh
at
s ign as a
E xa
I
a
b=d
e,
z
p ro
C o n i tio n s t
t
f
IM AG N A RY
AND
o
z
A q a ratic eq at io n c an n o t
C o n ditio n s fo r real, eq al, i
mf
i is ib le
fw
C HAPT E R
ud
u
Su
dv
igits is
n in e s
U RDS
i n ato r o
f a + ib
o
mpl V I I I
E xa
d
tities
ts O f u n i ty
ers o
o
s
fi
e rs o
C
o
.
t
are ro o
q
a
.
an
h
Mo
m f it
fa + Jb
o
+ i b = 0, t
a
su
f ( fit A g
/b
+ Jd
o
fa
o
mpl V I I I
I mgi
y qu
E xa
If
r
V III
are ro o
d th e
.
.
R atio n alis in g fac to r
u
u
an
g o u t th e
1
CH A PT E R
R ati o n ah s
Sq
b er
e en a n
AGE
P
.
Pro o f O f r le fo r
Te s t
um
w
iffe ren c e b e t
Th e
xi
.
0
.
m
”
bx
an
d
'2
a r
.
+b
'
x
c
’
0
vd
mhv
e re s o
ay
l
e
in to t wo
a e a co
mm
on
.
C O NTENT S
CHAPT E R X
u
vv
uk
u
m X
u
v vi w u k
u u
m
m X
u
v v v u
m X
d m u
s
l
e
p
E xa
in g o n e
ol
E q atio n s in
R ecip ro c al eq
atio n s
ples
.
gt
atio n
X (I
es
eral
.
atio n s
XI
m
ut ti
fp m
b
N um
b i ti
f m
b
N um
umb f mb i
Th
umb f mb i
er
N
b er
o
f
n atio n s
co
er o
um
mpl
w
Sign ifi eatio n
N
ic
m
g
n
,
,
titie s
mpl
eric al ex a
s
o
f
es
p
‘
71.
h
h
h
t in gs
.
a
e
n
’
an
d
e
‘
n
li
e
r at a
ti
e
h
v
e
IN A T I O N
u
m
dvd d
ti
u k
k
k
B
mi
r at a
c an
s e erally
t in gs
OM
S
.
e
t in gs
+p
C
AND
S
m
m
f 11 t in gs
,
m lik
.
f th e ter
o
an
s eq
i i
be
al
t o th e
e
in t o
’
m wh
um
h
h
m
k d h
dk d
lik
mu
um
h
m wh
hm
d
m
um
h
wh v u
d
mu
um
m
m
h
um
h wh
k
k d k
dk d
b er o f arran ge
are a
N
a
.
h
h
m+
wh h
in
ay s
XI
es
q
n
no
n atio n s o
co
clas s e s c o n tain in
E xa
titie s
f n t in gs r at a ti
o f n t in gs r at a ti
on s
na
er o
n
an
P ERM U TA T I O N
.
on s o
a
co
e n
q
n
e asy n
;
Preli in ary p ro p o s itio n
er o
w u
.
CHAPT E R
er o
i
n
u
t
ty
a
q
n
k w u
um
n
.
.
mpl
n o
.
s
in g se
c
p les
ete r in ate e q
E xa
n
o
n
.
ol
in
E xa
In
b
.
atio n s
Eq
w
a
.
E q atio n s in o l
H o o gen eo s e q
E xa
M I SC E LLA N E O U S E Q U AT I O N S
.
no
n
.
b er
o
e o
en
fo n e
f p er
ts
in
f n t in gs ta en all at a ti
q t in gs are ali e o f a s ec o n
o
,
tatio n s
o
f
71.
t in gs
r at a
ti
e,
e,
en
in
,
p t in gs
&c
en e ac
ay
be
rep eate
Th e t o tal
b er o f c o
b in atio n s
f n t in gs
”
TO fi n fo r
at
al e o f r th e e xp res s io n
C T is gre ate st
A b i n i ti o pro o f o f th e fo r
la fo r th e n
b er o f c o b in atio n s
n
t in gs
To tal
r at a
in
Of o n e
mpl
es
,
XI b
.
q
ali e o
o
fa
f 17 + q + r +
in
s ec o n
t in gs ,
,
&c
u
P du
ro
fp
.
m
o
E xa p le s
no
n
XII
e
e
ons o
.
f
n
are ali e
CHAPT E R X II MAT H E M AT I CA L IN D U C T I O N
f th mth d f p
f
ti
i lf t
f th f m +
t f bi m
stra
c
ere o
.
.
I ll
o
e
b er o f selectio n s
n
E xa
ti
o
a
o
o
ac o rs o
ro o
e
or
a:
a
.
C O NT E NT S
CH A PT E R X III
E xp an s io n
mi
d
m
Th e
D
fi c en
c o ef
mi
.
mi l th
Of
s
E xp an s io n
f
o
eq
u di
i
es
ts
o
f
o
m
m XIV
no
c as e
in
wh h
ic
b
.
t fro
s tan
mth
e
b egin n in g
s
s eq
u
al
to
su
mf
O
c o ef
fi c ien
IN O M I A L T H E O REM
mf y i d
B
.
m
.
h
hm i
w b
.
is
e xpre s s io n
( 33 + y )
art c
on
”
arit
ly
an
e t c ally
al
c an
O f th e e xp an s io n
ter
lar c as es
pp ro xi
o
f th e
atio n s Ob
.
N
b er O f ter
N
b er o f co
ay s
o
d
en
o
v
fe
en
n
.
ex
in telligib le
e xp an
e
IN D E x
dd
e
wh
en
<1
x
b y th e b in o
mi l
a
f (1
f (1
ex p an s io n s o
tain e
d
b y th e b in o
m
.
u
n
s
th e
mi l th
a
b in atio n
ples
c
.
mi
G en eral ter
e o re
m
s o
v
i
i
t
s
e
o
p
mi
es
ltin o
m
f 71 t in gs r at a ti
.
XV
ex p an s io n
th e
n
a ra tio n al
mpl
fa
md
XV
.
e
o u
t
Of n
al
e , rep etitio n s
b ein g allo
o
M U LT IN O M I A L T H E O REM
.
f
(
wh
a
in teger
G en e ral ter
is
h
o
dm
mu mi
wd
e
.
th e
n
du
ex p an s io n
C H APT E R
E xa
ts
ANY
.
b
ples
"
1
in th e ex p an s io n o f ( + x )
eric ally gre ate s t ter
i en s io n s fo r
c ts o f r
b er o f o o gen eo s p ro
letters
xa
d
e o re
en eral
N
an
.
h m
m
G
P i u
m
A
m XI V
um
hm
N um
um
mi
um
m
XIV
E m
E xa
th e fi rs t
a s
f (1
t
th e
q
u
an
ex p an s io n
tity
o
f
(
a
+ bx +
2
0 33
3
+ Lr +
(
fii
E I N T E G RA L I N D EX
m
E ul er s p ro o f o f th e b in o ial t eo re
or
“
Gen eral ter o f th e exp an s io n o f ( 1 + x )
E xa ple s
a
Th e
V
in te ge r
e
po n
mi
dd ter
’
o
S
m
CHAPT E R X I V
E xp an s io n
PO I T I
.
multi mi l
mpl XIII
E xa
s
e o re
o
fi cien
c o ef
er
t er
e p en
f th e greate s t ter
c o ef
fi cien t s
e
o
a
v
du
p o s iti
a
d
to
al .
m f th
mf
t m
o
e
m
.
ts
n atio n
ete r
Su
a
e
ity
n
XIII
i
u
are e q
Su
a
p ro o f o f th e b in o
a
p les
Seco n
E xa
u
s
an s o n
e ex
o
is
en n
m f th p i
my b md
i
exp an s o n
ter
IN O MI A L T H E O RE M
wh
f
o
G en eral ter
Th e
B
.
x
.
en
p is
wh
en
a
n
.
C O NT E NT S
CH APT E R X V I
D
efi n
itio n
m t y p p iti
E
mpl XVI
C m
m L g ithm
D t mi ti f th h
f l g ith mt
A dv t g
E le
es
xa
e er
na
dv
Gv
A
an
on
tage s
.
s
ar
o
o
w k
hm
ay s
th e lo garit
to b as e b
en
lo ga b
lo gb a
x
e ep in
f
s o
m XV I
b
.
o fa
mit
is th e li
E xp an s io n
ers
o
o
.
E X PON E N I I A L
.
wh
i)
1
hm
,
)
o
is in fi n ite
en n
a
t
f Tab les
ples
hm
f L o garit
s
mm
u
1)
lo g,
n
.
C HAPTE R X V III INTERE T
mu
v m m
P
V u
D u
v m m
mu
v m mu d
m
u u
mu d
v mm
P
V u
D u
v m mu d
m XV I I I
u
D
mu
u d u m
mu
u d u mu d
P
v u
u mu d
um
u h
P
v u d d u mu d
F
w
E m
XVII I
an
al
t
res e n
I n teres t
an
in al
No
C ase
o
re sen
E xa
f co
al
t
An n
A
o
n
t
o
A
o
n
t
o
N
t
d
e an
o
d tr
po
e
n
o
dA
an
ples
itie s
re s e n
dA
n
t
o
en
o
en
a
f
n p ai
an n
ity ,
si
f
n p ai
an n
ity ,
co
f an
rc
ity ,
o
n
en
at c o
t
po
n
.
co
p le in te re s t
p o n in tere s t
po
n
ity ,
co
in tere s t
as e
eferre
al o
b
po
p le in teres t
in te re s t
s
an n
b er o f y ears p
res en t
al e o f a
.
at si
su
.
itio n
ren e
A N N U ITI E s
p le in teres t
f a gi en su
at c o
al rate s o f in tere s t
efi n
in e fo r th e
f a gi
at Si
in teres t p ay ab le e ery
an d
is c o n t o f a gi en s u
.
.
su
n
al e o
ples
o
e an n
’
xa
f a gi
is c o n t
t
AND
S
.
I n teres t
s
L O GA RI T H MI C SERIE S
AND
n
—
f
f lo g, ( 1
ctio n o
'‘
R api ly co n ergin g s erie s fo r lo g, (n
Th e q an tity 3 is in c o
en s rab le
E xa
v
tis s a p o s iti e
to b ase a , to fi n d th e lo garit
Series fo r e
”
u
d v
u
m XV I I
C o n s tr
all n
an
.
CH A PT E R X V II
e
m
umb
g th e
1
E xa ples
E xp an s io n
b as e 1 0
o
s
ar
f al
b y in sp e c tio n
arac te ris tic
e c
o
a es o
an
i
o
on
o
a
.
.
ons
os
ro
ar
en
L O GARI T H M S
.
a10 ga N
N =
.
.
an n
f a leas e
po
n
in tere s t
.
in teres t
.
.
.
C O NT E NT S
C HAPT E R
m t y P p iti
A ith mti m
f tw p
m
Th
m f tw qu titi
E le
en
ar
e
r
ro
on s
os
c
ean o
o
o
o
X IX
v u
o s iti e
q
an
ean
e su
t
h
eq
Th e
en su
etic
mpl
a,
X IX
a
a
ax
a
an
tit
um
e an
,
en
,
mi im
m
mpw
f th
th
mpw
d
n
th e
p ro
su
e
q
mt i
e r c
wh
is greate s t
is leas t
e n t ey
an
al
th e ge o
an
ct
wh h
are
h
tities is greater t
e o
en
an
f
a
e
o
e rs
o
th
an
w
e
e ir
t
b er o f p o s iti
n
is greater
en
l e s b et
p o s iti
en
h
.
o
o
er o
f
a
0 an d 1
e en
in tegers ,
an
d
(
> b,
a
h
f t
n
um
b er
hm m
eir arit
v
f p o s iti e
e tic
ean ,
o
1
bb
mpl
E xa
es
XIX
b
.
C H APT E R
D
.
es
are
titie s is gre ate r t
t o fi n d th e greate s t
a an
etic
db
an
.
fa
c,
,
hm m
u i
wh mi
v
ex cep t
If a
du
ean
e r c
o
arit
q
o
mt i m
mf b
f m im
es
Th e
ean
c as es o
E xa
p ro
I N E Q U AL I T IE S
.
v h
du
v
m
v u
v u
b ein g gi
c t b ein g gi
es
a
th e geo
E a sy
al :
ul
hm m
arit
v
Gi
u
ey are e q
an
XV
efi n itio n o
mit
Li
o
.
XX
L I M I T IN G
.
m
V
ALU E
fL i it
f
I S ao
wh
S AND
xi
en
V
ANI H I N G RACT I O N
F
S
S
.
s z e ro
m
k
uh
m
m
m dw h
w
m md
mm
h
w
k
uh
d
m
h
m d wi h
w
md
d
m v h
d m
M h d
mu
D u
u
u
m uli
u
u
ud
u
P u
rie s a + a x + a 2x
h
e
e
r
f
t
S
e
o
t
g ,
0
1
y
h
u
f
ll
i
t
e
s
o
a
r
t
a
e
e p le as e co
a e a s large a s
p
ay b e
it ; an
b y ta in g a: large en o g , an y ter
t at fo llo
h
o f all t at
t
e su
t
r
e
a
e p leas e c o
a e a s large as
p
By ta in g a:
ay b e
p
et
o
o
e ter
s s io n
is c
o
f
in in g th e li
liaritie s in th e
mpl
E xa
es
so
l tio n
0
an
in
is in g frac tio n s
th e s o l tio n o f
Si
ltan e o
s
o
a ratic e q
fq
atio n s
.
m
o f ter
Seri es
f
XX
CHAPT E R XX I
C as e
o
aritie s
p ec
e
so
it s
atio n s
eq
ec
f
an
it
e
re c e
all e n o
s
O
18 co n
s
v
C
.
ON
V
ER G E N C Y
v
alte rn ate ly p o s iti
ergen
o
g
t If L t
m
u1
6
I
3
11
e an
13
-
1
d
AND
h
an
OF S
ERIE S
.
v
23 0
1
2 32
n egati e
le s s t
D I V ER G EN C Y
C O NT E NT S
X VI
Co
mp
au xi h ary
Th e
wh
f 2 a”
ari s o n o
s eri e s
A pplic atio n to B in o
m
lo g n
L i i ts o f —
u
Pro d
n
-
es
is
Series is
Series is
XXI
v
v
v
.
co n
u
a
T
a
ergen
co
n
mpl
E xa
es
,
wh
um
eq
u
atio n
b
b er Of fac to rs
v s eries
en
mp l
m
it
I
o
a
Pro o f o f p rin c iple
E xa
ples
2
lo g
n
E
L
u n +1
ec o
U se
m[ %
u
n
”
uw
t1
-
1
) l ]
l
1
lo g n
s eries
=
f
f
0 h as
ud
ud
m
mi
o re
n
eter
n
e te r
U N D ET ERMINE D C O E FF I C IE N T S
.
n e
mi
n e
h
d
d
t
an n ro o
ts , it is
an
d
i
en
c o e ffic ie n
ts fo r fi n ite
c o e ffi c ie n
t s fo r in fi n ite
tity
s erie s
s e rie s
.
.
P A RT I AL FRACT I O N
S
o s itio n
in to p artial frac tio n s
o f p art ial frac tio n s in e xp an sio n s
mpl
es
Scale
o
XXI II
.
f relatio n
mf
u
u
m XX IV
o
a rec
en eratin
E xa
1
Z a cp (n )
s eri es
C H A PT E R
G
1
mp
E xa
Su
)2
)
t,
n
CH A PTE R XX III
D
ergen
”
l
o
n
( g )
71.
.
o
b
.
v
1
t if L i
x
.
co n
1
n
(
dw h
t if L i
f( )
XX I
m XXII
es
is
-
.
Pro o f o f p rin cip le
E xa
erie s
ic
is in fi n ite
en n
CHAPT E R XX II
I f th e
hm S
E xp o n en tial, L o garit
wh
in fi n ite
.
3p
mi
are
ergen
XXI
t
mp
v
c t o f two
n
— +
+
t if L i
xili ary s eries
co n
2 19
m
i l
na
1
—
E
E v”
.
ergen
co n
Serie s is
du
1
se rie s
liary
x
a
ergen
co n
Serie s 2 ¢ ( n )
Pro
1
f an in fi n ite
ct o
s eries
Th e
an a
n
mpl
E xa
d
an
ui
it
.
ples
gf
rrin
g series
n c tio n
.
XX I V
RE CU RRIN G SERIE S
.
.
.
C O NT E NT S
u
u
So l tio n
G
f
o
u
f
o
Ny
x
2
1
2
2
n
y
Ny
2
Of x
l tio n
en eral s o
So l tio n
x
2
2
?
1
a
C HAPT E R XX I X
v u mh d
m
Su m
P
du
du
Su
M h d
f p re io
ary o
ct o
th e p ro
an
et
et
s
f n fac to rs in A
E x p res s io n
o
.
.
U M MAT I O N
OF S
ERIE S
.
.
ct o
o
o
S
s
o
re c ipro c al f th e p ro
b trac tio n
of
th e
an
.
f n fac to rs in A P
.
.
m ff t
b
t N m
f an
ac o rials
o
as s u
Po ly go n al an d Figu ra
ers
u
e
Pasc al s Trian gle
’
mpl
XXIX
hd D
h d u d wh
E xa
es
Me t
Met
o
ccee
s
s
is a ratio n al
I f an
.
ifferen ce s
f
o
o
a
.
a ratio n al
is
en a n
u
in tegral f
n c tio n
o
u
in tegral f
fn ,
n c tio n
o
fn
"
th e s erie s Z an e is a
re c
u
rrin
g
s erie s
uh
u
m XXIX
u mh d
M
S m
u um
B
m XXIX
F rt
er c ases o
E xa
l
s
e
p
.
is c ellan e o
lli
e rn o
et
s
’
N
s
o
fs
s o
umm
atio n
+
.
0
m t f p i ipl
f p im i i
b
N um
en
No
A
n
um
N
re s o
vd
l
mul
fo r
dv
v
w
dv f v
w
m
uv
h m
XXX
m
ay s an
in teger c an b e
i is o rs
o
f th e
es t
po
mt
mpl
Fer
E xa
efi n
fr
ct o
a
’
s
er o
T
es
itio n
c o n sec
.
o
fa
a
f a p ri
N
eo re
a
OF
N U M B ER S
ti
f c o n gru en t
en
e co n
e
p —l
.
gi
vd
re s o l e
in teger
tain e
d
dv
) wh
in tegers is
1 = M (p
in
m
t p ri
in to p ri e fac to rs in
gi en in teger
e
b er o f
o
H
a c a n re p re s en
o
H ig
D
b er c an b e
T E O RY
.
fi n ite
b er o f i is o rs
u
ro
n
s
es
r
XXX
.
es
ra tio n al algeb raic al
um
um
S m
h
P du
N
rn c
o
er o
”
.
C HAPT E R
State
n
b ers
p le s
E xa
.
f s erie s
o
u
b
g s erie s
rrin
f rec
es o n
on
ly
ly
on e
way
in to two facto rs
|n
i i sib le b y
ere
E
m
p is p ri
e an
d N p ri
mt
e
o
1)
C O NT E NTS
If
m
h
d
is p ri e to b , t
iffe ren t re ain
a
d
m
en
2a , 3 a ,
a,
i
x x
.
1)
(b
wh d v d d
a
i i
en
v
b le a
e
AGE
P
e
ers
(a ) «A ( b ) «n o
(N )
1
1
h m
u
h m(
du
m XXX
Wils o n
’
T
s
e o re
1
:
)
5
l
p
wh
(p )
1: l
m um
ere
i
p s
a
m
p ri
e
A p ro p erty pe c liar t o p ri e n
b ers
’
W ils o n s T e o re
s ec o n
p ro o f)
c tio n
Pro o fs b y in
p le s
E xa
b
.
d
.
CH A PT E R XXX I
ENE RA L
F RA C T I O NS
T H E O RY
G
TH E
.
C
OF
ONT I N U ED
.
mti
L aw o f fo r
b1
b2
+
“1
( 12
Th e
h as
v
d
v u
wh
G en eral
a
b2
+
1
al
a
mg
expre s se
ergen
ples
b
f co n
rab le ,
ud
ud
tin
as c o n
.
en
en s
ers io n Of o n e c o n tin
E xa
if
t
e o
a.
.
are
e,
al
e rgen
ts
mb
a n an
+1
itio n s
mpl
mp u
an
n+ 1
e
p o s i t1 ve p ro p er frac tI o n s
1
an
an
d bn
t
c an
ergen
if
§
fl
are c o n s tan
b e fo
ud
t
n
a
frac tio n
s
h
er
.
u
d ill
s tratio n s
mpl
Si
.
e
E
PR O A
B
.
v
en
B
ILITY
.
ts
XXXI I
d v
h w d d v w hh
P b b
mu h d
d d v
h
v wh h h
mu u
uvw
m XXXII
m
h
v h
v u
es
xa
Co
o
ro
a
n
C
E
ility t
la
Th e fo r
an c e o
f an
ple s
C an c e o f an
en
at
.
ts
t
in
o
ep en
o l s als o
e
E xa
en
.
e en
ic
t
b
t
fo r
m
an
en
t
ep e n
c an
e en
en
t
ap p en
ill b o t
ts
e en
in
ap p en
ap p e n in
g
al
e x ac tly r
t
e
ti
is p p
’
ts
ally e x c l s i e
.
d p ro b ab le
”
o f p o in t s
E x p ec tatio n
Pro b le
a
.
as ce n
<1
frac tio n in to an o t
e
0
bn
C H APT E R XXX II
efi n
>0
0
ud
v
wh
v u
v
mm u
gen eral
XXXI
d
v
m XXX I
2
a2
it
v
al e I f L t
b
n
a
I s In co
v u
efi n i te
f co n
es
{Series
v
c c es s i e c o n
?
+
Exa
Co n
e o
f
mpl
2
u
a
er o
ere
bl
d
a
ts t o
ergen
in g o r
fs
o
+
co n
C as es
on
a
es
in
n
trials
ay s
d
C O N T E NTS
X
E xan
In
v
X
XX
I
m
I
les
i
t
il
r
b
o
b
a
y
p
ers e
mt
State
c
.
u
mul
Pro o f o f fo r
m
P
p
( )
2
on
XXXII d
P
u m
m XXXII
E xa
es
L o cal
ro b ab ility .
.
.
M isc ellan e o
eo re
on
Tra itio n ary t es
mpl
t
my
tim y
C o n cu rren t tes ti
d
’
Q,
a
h
lli s T
o f B ern o
en
.
Geo
e r cal
e
p le s
s ex a
E xa ples
mth
mt i
e
.
o
d
s
.
CHAPT E R XXX III D E T ERM I N A N T
u li qu ti
i
t f tw h mg
E lim
i
h mg u li q u ti
E lim
t f th
D t mi t i t lt d b y i t h gi g w d lum
D v l pm t f d t mi t f thi d d
i
lt
d b y i t h gi g tw dj t
t i
f d t m
Sig
S
.
n an
o
n an
o
e er
n an
o
e er
a
o
o
n e ar e
s
en eo
o
a
s no
en
e e o
n
ree
n
e re
e er
n an
o
n an
s a
e re
on s
a
n ear e
s
en e o
o
o
o
a
o ns
ro
n
an
erc
.
ns
co
s an
o r er
r
e rc
n
n
an
o
a
ace n
um
w
m
w
d
d m v h
mm
um m
d u d
w
m
wh
u
md
um
du
d m
w
um
mm
P du
w d
m
m XXXIII
u
mu
u u
D m
uh d
D m
d
m XXXIII
co l
n s
o ro
s o r c o lu
If t
A fac to r c o
Cas e s
to
an
y
ere c o n s tit
en
ts
on
c tio n o
Re
ct o
ro
f
ft
en
are
ples
A p plic atio n to
a
.
tio n
f fo
rt
in an t
o
eter
in an t
O f an
ples
.
si
ay
fa
n
ifi c atio n
an is
b e p la ce o ts i
b e r o f te r s
o
f ro
s o r co
l
o
f Si
ltan e o
a tio n s
s eq
tien t
et
H
i i
en
5
0
o
f th e f
o
o
’
o rn er s
y
E xa
o
f
f
n
b
.
a
en
T
e t ac
et
o
e tric al an
en
S
tal la
by
e
e
o
n s
er
.
o
e
.
C HAPT E R XXX I V MI C E LLA N E O U T E O REM
vw
udm w
v m d f( )
f ( ) wh d v d d x
dvd d x
Qu
f ( ) wh
M h d
D hd
M h d
h Dv
m
S m
Fu
m
d
w kd
u ul mu
R e ie
es
o r er
y or
a1 b 2 c d
3 4
N o tatio n 2
n
in an t
e te r
.
so l
eter
l
e up o
a
in an ts b y
e ter in an ts
o
tic al, th e
o r co
ro
ete r
E xa
E xa
i
n s are
ro
a
lea
i i
e
s o
f A lgeb ra
e s re
ain
by
a
C o e ffic ien ts
f Sy n t
e tic
d Altern atin g
ple s o f i en titie s
L is t o f s ef fo r
lae
or e
H
S
i is io n
n c tio n s
o ut
er
a
S AN D
w
s
mpl
E xa
d
XXXIV
vd
es
tities p ro
Lin ear fac to rs
I
en
Value o f a
a
.
u
b y p ro p ertie s O f c b e
3
3
3
Of a + b + c
3a b c
e
wh
+b +c
n
n
.
n
ro o t s
+b + c=0
en a
mpl XXXIV b
i ti
E lim
b y ym
mt i l f
i
E lim
ti
mth d f limi ti
E l
E xa
es
na
on
na
on
’
u er S
v
.
.
o
e
un c tio n s
e r ca
s
e
o
on
n a
Di M h d
mh d
u m
mi
m XXXI V
Sy l
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C H A PTE R
II
PR O POR TI ON
N
N
18
D E F I I TI O
t
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If fo u r qu a n ti ti es a re i n p rop o rti o n , th e p ro du ct
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I t w ill b e s e e n t a t t i p ro p o s it io n is t h e
li
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a
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m
21
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I f f u qu
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23
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ul s f th p i
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Th
ft qu t d b y th
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PR O PO R TI O N
1
( )
Fo r
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5 22
,
b
d, t
= c
t
h
h f
6
en
a
I
e re o re
Fo r a d
t
h
at
b
a
:
be
t
e
d, t
h
e
ad
h f
e re o re
Fo r
t h at
If
a
b
b
a
e
’
d
e
b
t h e re
f
a
o re
a +
b
+
b
z
Fo r
If
a
6
b
a
e
d
e
d, t
h
en
t
h f
a
e re o re
—
5
a
]
.
b
e
d
z
e
1
— —
b
c
]
D
viden da
i
[
]
d
b
b
a
da
o n en
5
d
e +
z
m
p
d
c +
d
b
o
C
[
d
e
r
1,
+
b
a
.
d
b
a
1
Is
4
( )
b
+
E
d
J
d
e
d, t h e n
b
e
5
62
C
3
( )
A
l
t
rn a n da
e
[
.
-
a
a
e
be
a
is
]
;
d
a
da
verten
’
d
a
en
l
n
[
.
0
d
a =
e
e
b
b
d
:
l
b
is
If
a
15
.
d
e
1;
61
d
0
b
a
5
( )
If
a
z
b
z
e z
—
b
d by
a +
a
4
( )
a +
T is pro po s it io n is
den do
—
d
b
z a
e
z
a +
b
b
z
d
—b
b
g s
b
c
a
b
e
d
.
+
;
d
d
—
u u lly qu
a
Se ve ra l o th e r pro po rt io n s
e
d
:a
.
d
d
a +
s
e +
:
d
c +
b
b y di v1 s 10 n ,
h
b
d, t h e n
Fo r b y ( 3 )
an
z
d
o
te d
my b
a
e
p
as
ro
v
d
Co
e
m
p
d in
o n en
do
an
mil
a si
ar
d D ivi
wa y
.
16
GH E R
HI
AL
u
G E BR A
.
Th e re s lt s O f t h e p re c e din g a rt ic le a re t h e a lge b ra ica l
24
t h b o o k O f E c lid,
e o f t h e p ro p o s it io n s in t h e
u iv a le n t s o f s o
q
i s el
ak e
a
ilia r
it t e
a n d t h e s t de n t is a d is e d t o
Fo r e x a p le , dividen do
a
as
b
e
o
d
in t e ir e rb a l o r
t
e
y
q
o llo
m
v
f m
.
u
h v
f ws
Wh en
fif
m hm f f m
th ere
a re
f
d is to th e s eco n d,
u rth
i
t
t
h
e
o
o
o
u
r
t
h
s
f
f
th e
the
ro o rtio n a ls ,
p p
ou r
th e
as
s eco n
w h hm
m
m
.
u
i
r
s
t
o
a
b
v
e
f
f
th e th ird a b o ve the
exc es s
f
o
ex c ess
u
the
o
.
h ll w mp t h
w i h t h t giv i E u lid
i
f ll w
E u lid s d fi i i
id
b p
F u qu i i
We
25
rt io n
o
p
.
t
c
en
a
’
t
n
e
co
n o
a
s
c
n
s as
on
e a l e b ra ic a l
a re
o
de
g
fi
n
it io n
o
f p ro
.
o
s
wh e n if a n y equ i
rs t a n d t ird, a n d a ls o a n
u lti les wh a te ver b e t a k e n o f t h e
y
p
o
rt , t h e
i
u lti les wh a tev er o f t h e s e c o n d a n d
l
t
l
e
o
f
e qu i
p
p
lt ip le o f t h e
t h e t h ird is gre a t e r t a n , e q a l t o , o r le s s t a n t h e
lt ip le o f t h e firs t is gre a te r t an , e q a l
rt , a c c o rdin g a s t h e
o
lt ip le o f t h e s e c o n d
t o , o r le s s t an t h e
m
an
r
o
h
In
F
a
lgeb ra ic a l
u qu
r
o
di n g
ac co r
as
an
p
to
e s a re s a
m
fu h
s
h
u
mu
mu
y mb l h
o
t itie s
a,
b
q , p
a
u
s
.
fi
t
b,
e
’
d
ro
t
q
b ei n g
hp v
II
ro
.
t h e ge o
es
n
p
G i ve n t h a t p e
I
e rt ie s o
z
yd
fi
O
de
fi
n
it io n
o
0
in
en
tegers
b
o th
g
f
f
s ide s
by
5
,
we
Ob
t a in
PC
9 05
ra c t io n s ,
di n g
acco r
as
p
a
b
,
g
:
.
fi
n
it io n
o
0
d
acco r i n
(T
f p ro po rt io n
g
as
p
a
gb , t o p ro
I
v
e
e
z qd
wha tever
f p ro p o rt io n
.
gd
u
s s a e
on
y p o s i ti ve
To de d c e t h e a lge b ra ic a l de
e t ric a l de n it io n
m
o r
an
e ric a l
t h e p ro po s it io n
u
ro
t
e
.
pe
wh ic
in
a
0
mt
96
ro
h
fu h
mu
h
mu
h
my b h u t t d
wh
t
i
p p
p
O
by
f m he p
fi
a re
a
p
h e n c e,
rt io n a ls
o
p
de fi n it io n
e
lt iply i
m
u
; d
Si n c e f
ro
.
e,
d
an
To de d c e t h e geo
a l e b ra ic al de n it io n
g
I
th e
t t
.
f m
ro
PR O POR TI O N
If
a
b
1
s
a
u
t
n o
qu
th
e
e
a
e
l to
on e
at
o
f t
17
.
h mmu t
e
s
b
th e
e
mf
>
i
S ppo s e
e
n
t
w
i
l
l b e p o s s ib le t o fi n d s o
;
b 3
w ic lie s b e t e e n t e , 9 a n d p b e in
s it i e in t e
o
gp
h h
H
w
hm
v
Q
en c e
e
g
.
ra c t io n
e rs
g
re a t e r
.
g
b
0
2
9
d
m( 1 )
f m( 2 )
FrO
p
an
d t h es e
co n
f g ;
i
i
s
t
p
I t h u ld b
d ls wit h
Th e re
i
an d
o re
o
a re n o
t
(
t h e p ro
26
rt io n
o
p
on
o
e
hyp t h
t radic t t h e
b
g ;
d
g ;
>
a
p
ro
p
u
e s is
n e
u
q
.
al
t h at is
wh ic
hp
ro ve
s
.
t ic e d t
h
t h e geo
mt
e ric a l
fi
de n it io n o f p ro
a n it de s
c o n c rete
ea
s
c
a s lin e s o r a re as
,
g
,
b
u t n o t re e rre d t o an
o
e t ric a ll
re re s e n t e d
e
co
o
n
n it
g
p
y
’
en t
So t a t E c lid de n it io n is a pplic a b le t o in
of
eas re
e n s ra b le
e n s rab le a s w e ll a s t o c o
co
an t it ie s ;
e re a s
q
t h e a lgeb ra ic al de n itio n , s t ric t l s p e akin g, a pplie s o n l t o c o
u
an t it ie s , s in c e it t a c it l
ass
e n s ra b le
es t h a t a is t h e s a
e
q
lt iple , p a rt , o r p a rt s , o f b t a t e is o f d
de t e r in at e
B u t th e
i
a e b e en
i
ra b le
ro o
w
h
ch
e n fo r c o
en
a
n t it ie s will
p
g
q
t ill b e t r e fo r in c o
e n s rab le s , s in c e t h e ra t io o f t wo in c o
a de t o di fe r ro
e n su rab le s c a n a l a s b e
t h e ra tio o f t wo
in t e ge r b y les s th a n a n y a ss ign a ble qu a n tity
T is h a s b ee n
a
ro ve d
re
a ls o b e
n in A rt 7 3 it
o
e
n e ra ll
a s in
sh e
t
h
e
p
y
g
n ex t a rt ic le
o
s
.
e n o
m
m u m
mm u
fi
m u
m mu
h
at
m
y
u
f
mm u
s fi
mm u
u
wh
y
y
m
y um
m
h
mm su
fs
hv
v
u
mm u
s
u
m
m
wy m
f f m
s
h
m
m
y
w
.
u
uh
.
.
.
.
27
u
u
h
mm u
v
h
m h m
m h
e n s ra b le ; di ide b
S pp o s e t a t a an d b a re in c o
in t o an e q al p a rt e ac h e q al t o B , s o t a t b
w
i
a
re
e
s
,
fi
l
so s
o s it ive in t e e r
A
o s e ,8 is c o n t a in e d in a
o re t an n
p
g
pp
1 ti es 3
t i e s a n d le ss t an n
.
s
m
u
.
h
E>
6
-
b
t
h
so
at
t
is ,
lie
s
m
n
m3
e rs
.
H
.
A
.
ro
n
+
1 ) ’B
mfl
an
d
”
,
l
l
a
7'
at
(
an d <
b e twe e n
h ; diff f m2mb y
H
u
a
u
q
an
t it y le s s t
h
an
A
n
d s in c e we
18
c an c
b
GH ER
H I
h
o o se
(
8
,
md
e
w ill
it
n
o
f
re
e as
en
H
.
t
)
.
smll
as
a
1
en ce
be
c an
m
we plea s e ,
as
771.
md
a
m
mll
c an
e as s
f u wh
a
o s e ra t io
be o nd
an
re
i
re d de re e o f a c c ra c
y
q
g
d t wo in t e ge rs
t a t o f a an d b t o
e x p re s s
GE B R A
m u m
we ple ase
as
g
w e ple a s e ,
as
u
re a t
e as
a
o u r
AL
an
an
n
h
v
u
d
u
c an
f
y
u
u
fu
.
Th e p ro p o s itio n s p ro e d in A rt 2 3 a re o t e n
l in
28
se
I n p a rt ic la r, t h e s o l t io n o f c e rt a in e q a
s
s o l in g p ro b le
a c ilit a t e d b
a s kil
l
u s e o f th e o
e ra t io n s c o
t io n s is gre at l
y
p
o
n en do a n d di vi den do
p
m
.
v
yf
.
.
fu
u
u
m
.
m
m mb
E x a p le 1
I f (2 a
6
.
p ro
e
co
b,
at a ,
t
mp
o n en
d
d
e,
2
a e
are
p ro p o rtio n als
b
3 710 9 n d
m 6m
a
2
m 6 mb
a
o an
3 ne
dvd d
d i i
en
o
.
9n d
m+ 3 )
2 ( 6m
b
9 d)
2m 3
2m 3
2 (2
A gain
an
,
d
co
o
,
mp
o n en
d
o an
d
a
ne
a
ne
dvd d
i i
en
4
m
,
12
(3n e
A gain ,
co
mp
o n en
d
o
an
d
d
b=
e
en
3 2x 2
2x
2
9n d
a
3n e )
9 nd)
9n d
9n d
mb
b
e
dvd d
i i
3 71 0
18nd
a
a
a
mb
6m
b
a
W h en c e
9n d
6
o
o
’
d
.
,
'
1 6x + 1 0
3 2x
-
8
1 6x — 8 x + 5
1 6x — 4
2
x
wh
en c e
1 6x
.
‘
3
— 4 x = 1 6 x 2 — 8x + 5
;
D
a
3n e
2 (2
ne
m 6 mb
a
m
2 (6 m
b
,
a
9 u d)
9 n d) ( 2
m 6a
2m 6 m
b
2
n
A ltern
3no
3n e
a
a
v h
W h v
e
m 6 mb
b
(2m 6 m
9 u d) ( 2
3ne
9 n d) ,
20
HI
GH E R
AL
G E BR A
.
m
k
d
w kd
ay s is t o th e
e n in x + 1
I f th e wo r do n e b y x 1
1 0, fin d x
1 day s in th e rat io o f 9
en in x
by x +2
18
m
.
F
.
dfu
uh h
t
in
o r p ro p o rt io n als s c
19
o f th e
ean s 1 9 , an d th e
2 1 , th e s u
b ers is 4 4 2
n
.
um
one
or
m
m
at
su
th e
m f th
su
m f th
o
o
e sq
u
e e x tre
m
is
es
fu
ares o f all
o
r
.
Bw fi d h k d h m
k
B
h u
k f m h
mu
m m
h h h
k d
h
i
f
l
l
f
it i t h
f
k
u
w
l
d
w
m
l
Ni
21
g
fill d wi h w the i g ll f th mi u
d w d th
k i g i fill d with w t I f h qu i y f wi w i h k
h w m
u h d th k
tity f w t
i it
b
th q u
6 t
h ld 1
i
hw
ti u d p p t i
If f u p itiv q u ti i
22
h fi
b w
dl
h im
t h t h d iff
i
t l
h diff
b w
th
h w
t
g
I E gl d th p p u l i
i
w
d
b
23
p
if h t w p p u l i
i
d
d
d h
7
p
mp h w d u y p p l
u y p pul i p
Two c as ks A an d
e re
lle wit two in s o f s erry , ix ed
in t h e ratio o f
A in th e rat io o f 2
in
c as
7 , an d in th e c as k
ixt u re
eac
t o fo r
5
s t b e t a en
ro
a
1
W at q an tity
w ic s all c o n s is t o f 2 gallo n s o f o n e in an d 9 gallo n s o f th e o t er ?
20
th e
.
.
ne
.
t
e
e
a n
to
ons
a er
n
xt
e
t t
9,
e
an
as
1
o
.
o
o
t
a er
e
o
ne
re are
ra
a c as
ro
n
a
n ne
n
an
e
ra
o n s are
at er,
s a
cas
a
n
ne no
o
en
,
an
e
t
n
e cas
o es
c
o
s
;
e c as
o
t
a
t
as
co
n
n
n
tr
t io n s in 1 8 7 1
24
co n s
co
ee
c en
w
t
m
e re
m
co
on
een
e
o
o
n
4
rs t an
e
e o
t
at o n
t
,
co n
as t
er
t
o
n
eas t
s a
co
are
or
ro
e
t
on
,
t
ree
s
e
e s as
.
1 59
n c reas e
18
t e t o n an
n c reas e
at o n
o
er c en
n
t
cen t
n tr
er c en
er
co
et
.
.
een
t
an
o
e
u a
th e
umpti
m
mu
m
m
f t e a is fi ve t i es th e
o f co f
I f a pe r c en t
fee
o re t e a an d b p e r c e n t
o re
co n s
ed, t h e aggregat e a
o
n t co n s
o
e
ld b e 7 0 p er
o re tea an d a p e r c e n t
o re c o f
fee
; b u t if 6 p er ce n t
ed, th e a gre at e a
n
c
n
t
o
o
s
u
e
o
l
b
e 3 c p e r ce n t
g
g
p are a an d b
um
c o u ntry
co ns
.
on
m
mu
o
u md w u
m
md w u d
.
.
.
.
.
.
Bras s is
25
et
e
m
e re c o n s
o re
t
certain
um
o re
.
t
e s are
.
In a
t
i
n
o
p
um
ff w
.
an
1 88 1 ;
o
at
an
e en
eren ce
e
,
18 1
et
t
an
e
os
eren c e
e
rea
r
o
.
allo y
d z in c ; b ro n z e is an allo y
c o n tain in g 80 p er c en t o f c o
r
4
o
n
n
d
1
t
i
n
f
z
i
c
a
o
f
A
f
u
s
e
e
6
,
,
pp
as s o f b ras s an d b ro n z e is fo n
t o c o n ta in 7 4 p e r c en t o f c o p p e r, 1 6
o f z in c , an d 1 0 o f t in
fi n d th e ratio o f c o ppe r to z in c in th e co po s itio n
o f b ras s
.
an
m
o
f
c o pp er an
ud
.
d
.
m
.
.
26
A
c re
w
c an
ro
w
a c ertain
w h
m u
hy
h uld w i i ill w
wi h h
m?
t
t
e
.
c an
ey c o
t
t
ro
ro
e s t rea
t
e
t
sa
n st
e co
rs e
at e r :
co
dw
o
n
u
rs e
s trea
h o w lo n g
s tre a
mi
up
9
n
mi 84 m
i ut
mi u t l th
n
o
t
es
n
w uld h y k
e
ta
es
n
e
to
es s
ro
w
an
dw
o
;
n
C H A PTE R
I II
VA R IA TI ON
N
N
u
.
.
y
On e q a n tit A is s a id t o vary direct ly
D E F I ITIO
a n t it ie s de e n d
a s a n o t h e r B , w en t h e t wo
o
n
a
e
h
c
o t e r in
q
p
p
s ch a
an n e r t a t if B is c h an
d
i
e
A
s
h
an
c
e d in th e s a
e
,
g
g
ra tio
29
.
h
m
u
.
u
h
.
as
N OTE
B
w d
Th e
.
directly is
or
o fte n
h
u
o
mitt d
e
an
,
m
dt v
d A is
s ai
o
ar
y
.
Fo r in s t an c e
mv
u
f m
v
o in
if a t ra in
at a
n i o r
ra t e t ra e ls
g
in t e , it will t ra e l 2 0
iles in 3 0
ile in 6 0
in t e s ,
40
in t e s , a n d s o o n ; t h e dis t an c e in e a c h c ase
ile in 1 2 0
80
b e in g in c re a s e d o r di in is e d in t h e a e ra t io a s t h e t i e
en th e
n i or
e lo c it
is
T i is e x p re s s e d b y s a y in g t at
th e dis ta n c e i s p rop o rtio n a l to the ti e, o r th e dis ta n c e va ri es a s
th e ti e
:
mu s
mu
m h
ms
ms
hs
m
sm
v
h wh
v
m
m
mu
y
m
f m
.
u
.
30
Th e s
A at B is re a d
-
.
31
If A
.
c o n s ta n
v
u
l
es o
h
T
H
o
“
s
u pp o
fA
en
,
se
an
en c e
h
v
de n o t e
to
”
i
a r a t io n
;
t
so
h
at
.
B , then A is
dB
at a ,
equ a l
to B
mltip li d b y m
u
e
so
dl
,
b, b ,, b2,
“
2,
a re c o rre s
n
o
p
.
b y de fi n i ti o n ,
al
a
bl
b,
v
y
B
b
a
a
u
e o
2
fA
,
a
is
e o
5
a
w
l
a
=
s
.
an
d
so
3
3
u
y
i s c o n stan
e re
A
“
b el n g e q
v lu f B
m wh m
di n g
b
a
b,
“
ea c h
63
al
b
a
b;
fl,
,
A
I s,
B
s ed
e
.
c o rre s p o n
th at
i
ar es as
t
an
u
is
cc
va ri es a s
t qu a n ti ty
Fo r
a
y mb l
A v
al
s
t
.
to
th e
sa
m
e
on
,
din g
22
HI
If
an
i
y p
a r o
th e
c o n s ta n
B
1a
=
we h a
v
m
t
f
GH E R
be
G E BR A
v lu s
din g
det e r in e d
c o rre s o n
p
can
AL
m
o
e
a
.
Fo r
.
f A an d B a re k n o wn ,
in s ta n c e , if A
3 wh e n
e
A
i
i
i
N
u
t
t
d
O
ly
y i
y
q
h
wh A v i di ly th ip l f B
m
wh
ly
B A
i
s
Th u s if A v i s i v
m
t
t
B
illu
ti
fi v
v i ti I f 6 m
Th f ll w i g i
i w k i 8 h u
1 2 m w u ld d t h
mw k i
d
Thu it pp
d
s th t
2 m i 24 h
4 h u
;
w h t h u mb f m i i
d t h t im i p p ti
t ly
a
d;
d vi v
d
T
u
v iv
h
u
m
wh
wh
§
om
o f/
o
h m
w h v
Pu i
D E FI
a n o t e r B,
32
as
I TI O N
.
ar e
a c e rt a n
o
en
ce
an
e c re a se
he
E x a p le 1
en y = 3 , fi n d
x = 8
en
at
!
s u pp o s
tt
e n ce ,
on
on
e re
e rs e
ar a
e
,
e
a
s
an
on
e sa
s
.
.
s co n
9
o
o
ro
.
en
e
or
n
ea r
or
a
on a e
aries
Of x
t
=
1
ers ely
n
as
t
e s q are o
f y ; if
.
1 8 c o n s tan
e re
t
.
31
8, y = 3 ,
ngx=
u
2
a e
e
we
b y p ttin g y
m
u
um
m
y
W
x
so
ro c a
n vers e
.
ro o
”I
By
h
be
c
.
o
n c re a s e
s
e rs
-
n
en
an
o var
sa
e re c
?
o
rs ,
en
as
,
on
o u rs
n
er o
e n
en
o
n
or
rs ,
o
s t ra
s an
n
o
o
e
as
e rs e
n
s
rec t
ar es
en
t
an
n e
.
u
o b tain x
= 51 2
.
m
v u v
V u
v u
Th e sq are o f th e ti e o f a p lan e t s re o l tio n
a rie s a s
E x a p le 2
’
th e S u n ; fi n d th e ti e o f en s re o l tio n ,
th e c b e o f its is tan c e fro
in g th e di s tan ces o f th e E art an d e n s fro
th e Su n t o b e 9 1 } an d
as s
illio n s o f ile s re sp e c ti ely
66
.
d
h V u
v
mm u d d
m
d
hv
or
wh
e re
k is
so
h
wh
,
t
in
re
ea s
P
g
D
oc
P
e co n s ta n
Fo r th e E art
e
”:
m
m
m
.
L et P b e th e p erio ic ti
iles ; we a e
of
m
m
’
ay s ,
D th e
3
kD
,
3
,
.
3 6 5x 3 6 5: k x 9 1 3; x 9 1 1 x 9 1 3
1,
4
x
e n ce
4
x
4
.
365
4
x
4
x
365
4
d
is tan c e in
milli
ons
V A R I A TI O N
u
Fo r V en
wh
s,
x 66 x 6 6
66
x
3
.
en ce
x J 7 2 33
2 64
'
,
mt ly
ap p ro xi
a e
,
—2 6 4 x 8 5
H
m
th e ti
en c e
v u
f re
e o
N
o
l tio n is
N
w
v i s j i tly
d
n early
u
y
ay s
.
33
D E F I I TI O
On e q a n t it is s a id t o vary j o in t ly
b e r o f o t h e rs ,
h e n it v a rie s dire c tl a s t h e ir p ro d c t
um
n
.
Th
us
p
p
34
o n
ar e
t h e in t e re s t
s tan c e ,
rin c i
A
.
m
a l,
th e t i
D E FI
.
on
N
N
v
0, wh e n A
v e rs e ly a s
ra t e
A is
.
h
m y v
r cen
e
p
s a id
i
mB 0
A =
a rie s
j
.
.
y
o in
tl
as
B
Fo r in
th e
as
.
y
dire c tl
ar
an
d in
C
3 5 If A va ries a s B wh en C is
wh en B is co n s tan t, th en will A va ry
.
v
t
en
v y
to
B
a r es a s
u
d 0, w
of
on e
an
m
su
d th e
e, an
ITI O
a
B
as
y
as a
d A va ries a s C
B C wh en b o th B a n d C
co n s ta n
as
y
t,
an
h
y
Th e a riat io n o f A dep en ds partl o n t a t o f B an d p a rt l o n
S pp o s e t h e s e la t t e r a ria t io n s t o t a k e pla c e s e pa
th at o f 0
f
e c t o n A ; a ls o le t
ra t e l , e a c h in it s t rn p ro d c in g it s o wn e f
y
lt an e o
v al e o f A , B , C
a , b , c b e c e rt a in s i
u
.
u
u
us
w h il
u
mu
1
L et C b e
a rt ia l
n de r o a
g
p
’
h e re
a ,
.
w
c o n s ta n
c
h
an
t
h
e an
g
d will
A
2
an
B
e
a
c
v
us
L et B b e
.
s
e
g
to
mdi
in te r
e
e
a te
t
h
co n s ta n
en
v lu
a
A
e a
’
fi
t o it s
an
d ( 2)
'
e so
'
A
d
’
l
v lu
a
it s
a
c
e a,
B
C
b
c
a
9
t h a t is ,
be
v
n
i
.
an
t
h
en
mdi
ter
BC
.
v lu
e an
e re
B C,
a r es a s
it s
h g
wh
C
x
A
re t a in
c
’
e
b
A
mu t
s
u
a t e va l e
e
b
e e
n a
g
to
B
co
s
es
um m i
a ss
mu t mpl t
a
ro
c h an
t, t h at is , le t it
a
F m( 1 )
.
a
e
b,
d pa s s
wh il 0
f mit
e
ro
s
24
H I
GH E R
AL
G E BR A
.
f w i g illu t ti s f th t h mp v d i
th
mu f w k d b y gi
mb of m
Th
i s
u mb f d y t h y w k d th mu t f
di
t ly
th
w k d i gi im v i di ly t h u mb f m ;
f wh h u mb f d y d t h u mb f m
th
h p du t f
mu t f w k w ill v y
b th v i b l th
u mb f m d t h umb f d y
th
i gl v i s di
f
tly
A g i i G mt y t h
t
t
d di
tly
th
h igh t
it b
wh
th
h igh t i
b t h t h h igh t
wh th b i
t
t;
d wh
d b
h
d
t
f
t
v i b l th
v i
u
u
th
m
b
p
i
i
h
h
t
d
h
b
h
t
g
p
g
v um
h u
v
u
m
du
wh
h h
h h wh
du
h h
v um
u
h h
wh v m
u
wh h d
wh d u
d
v
h h d du
b
m u d
v um u
h
m wh m
u
m x
wh
Th e o llo
36
la s t a rt ic le
e
e
o
eo re
ro
e
n
.
n
o
a
e
on e
or
n
a n
t
en
s
as e
e
e,
ar a
re s e n
t
n
s
c o n s an
e
e
as
ar e s
t
an
ase
e
er o
t
as
an
e
n
o
en
en
ro
a re
c
o
as
re c
e
ro
re c
ar e
e
o
e
o
.
an
,
en
an
a re a
tr
a
c o n s an
s
a
a
er o
e n
ar
o
e
e n
an
er o
a re a
e
s
va r e
en
an
,
as
or
e n
e
e
a
er
n u
or
re c t
o
e r
e
e
ar es
n
an
eo
en
ase
a re
en
s
er o
o
a
ven
a
a
e
e n
e
n
,
t
ven
a
er o
e n
on e
or
er o
e,
ar a
s
o
n
en
e re o re
o
t
e
as
rec
re
on
s ra
a re
n
.
e
e
o
c
e
as
e
an
as e
n
e rs
.
arie s as th e s q are o f th e
E x a p le
Th e o l
e o f a rig t c irc lar co n e
e ig t is co n s t an t, a n d as th e
ra i s O f th e b ase
en th e
e ig t
e n th e
I f th e ra i s o f th e b as e is 7 feet an d th e e ig t 1 5 feet ,
b as e is c o n s tan t
e ig t o f a c o n e
o se
th e o l
e is 7 7 0 c b ic fe e t ; fi n d th e
o lu
e is 1 32
ic s tan s o n a b as e
c b ic fe e t an d
o s e ra i s is 3 fe et
.
.
.
L et h
eas
re
T
en
By
d r
fe e t ;
an
in
V
2
r
:
en o
als o
h,
pp o s itio n
s
te res p ec ti ely t h e
eig t an
le t V b e th e o l
e in c b ic fee t
ra
i
s o
f th e
as e
.
ere
is
co n s tan
7 7 0=
,
t
.
x
2
7
15
22
en c e
22
21
by
s
u
u
b s tit tin g V
132,
:
r_
we get
22
—
21
wh
an
en c e
dt
h
ere fo re
th e
h h
eig
X
9
X
h
,
h : 14 ;
t is 1 4 feet
.
y
37
Th e p ro p o s it io n o f A rt 3 5 c a n e a s il b e e x t e n de d t o t h e
c a s e in wh ic
t h e a ria t io n o f A dep e n ds p o n t a t o f
o re t a n
t wo
a ria b le s
rt e r, t h e
a ria t io n s
a
i
b
e
e
t
e r dire c t o r
y
in e rs e
Th e p rin c iple is in t e re s t in g b e c a s e o f it s re q en t o c
c rre n c e in
P y s ic a l Sc ie n c e
Fo r ex a ple, in t h e t e o r o f
a s e s it is
o
n
d
b
x
r
i
n
a
h
e
e
e
t
t
t
t
e
re s s re
o
f
a
as
g
y
p
p
p)
(
g
“
va ri e s a s t h e
ab s o l t e t e
ra t re
w
e
t
n
i
l
u
e
t
s
o
e
v
i
s
p
()
( )
c o n s ta n t , a n d t a t t h e
re s s re
a rie
i
n
e
r
s
l
a s th e
e
o
l
e
p
w en t h e t e p e ra t re is c o n st a n t t at is
.
v
u
h
h
v
.
.
v
Fu h
h
fu
m
.
u
m
u
m
v
m h
m u
.
h
u
u v
u
p
or
t,
wh
en
v
is
h
h
h
m
h
f u
h y
u
h
v m
s v y
v um
c o n s t an
t;
V A R IA TI O N
1
25
.
i
wh
5
F mt h
u lt w h u ld p t t h t wh b h d
v i b l w h uld h v t h f mu l
h
wh
I
t
t
p
p
5
d by
tu l
i
m
t
t
h
i
i
f
u
d t b th
p
m
du
w ju v d
d
v
v
v
v d
u
u
u d m
v
um
h
hu
m
j u
u d w mu h
w
w
w h
umd j u
m
mu w h
m
di h u
d
m
v
m
hu
u
w
um
W h v
ro
ese
ar a
e,
re s
e s
s
o
e s
t
an
ac
a
ex
or
e
v=
or
a
.
en
,
o
t
t
an
v
t I s c o n s an
e re
s
t
a
t,
en
er
ec
ex
e
a
cc
o
s c o n s t an
t
en
s
n
o
o
e
e c a se
.
ratio n
ire c tly a s th e
Th e
o f a ra il ay
o rn ey
E x a p le
arie s
is tan c e an d in e rs ely a s th e elo c ity ; th e elo c ity arie s ire c tly as th e
se
ile , an d in e rs e ly a s th e
s q are ro o t o f th e q an tity o f c o al
p er
I n a o rn ey o f 2 5 ile s in
a lf a n
n
b er o f c a rriage s in th e t rain
o
r
c
c o al
ill b e
it
1 8 c arriage s 1 0 c t o f c o al is req ire ; h o
in te s it 1 6 carriages ?
i n a o rn ey o f 2 1
iles in 28
co n s
e
.
.
.
L e t t b e th e ti e e xp re s s e
n
o
d th e is tan ce in
ile s ,
v th e
e lo c ity in
ile s p er o
q th e q an tity o f c o al in c t ,
b er o f c arriage s
e th e n
rs ,
r,
.
.
e
a e
9
C
wh
ed
en ce
da
_
u
u
S b s tit tin g th e
v u
wh
ke d
_
_
Ja
al e s
v
gi
en
,
we
1
k is
e re
c o n s tan
t
.
hv
a e
x 25
k x 18
2
h
t
at I s ,
H
q
u
_
25
en ce
u
u
S b s tit tin g
e s t io n
,
we
hv
no
w th e
a e
v u
al
es
28
60
h
wh
t
at
Jfi
J
is ,
x 36
1
J 0
ed
2 5x 3 6
Jq
o
f t,
Jfi
e,
2 5x 3 6
x
1 5x 2 8
32
en c e
H
en ce
3
th e q
u
an
tity
o
f c o al is 6 -3 c wt
en
zr
i ex f
x
10 x 1 6
v
d gi
.
Jq
21
_
4
3
in
s ec o n
d
26
HI
GH E R
AL
GE BR A
E XA M PLES
1
.
2
Q
=
If P
.
I f th e s q
.
u
A v
fin d t
A
en
hv
v
If A
5
.
54
=
ary as
0
v
Q,
e rs ely as
en
an
varie s as
fx
en x
d P= 7
wh
ub
fy ,
th e
aries as
j o in tly ;
d C
an
0,
1 5, fi n d
=
y
.
c
e o
h
w
x
Q
en
=
en
10
=
y
.
h
3 , fin d P w
h
dx= 3 w
an
en
en
=
y
4,
JS
dB= 3
an
wh
d x= 8
1
h
B
w
arie s as
.
v
are o
val e o fy
fin d th e
4
u
an
y,
in
varies
2%
3
e ac
varies as
If x
III
.
.
wh
2
B
en
§ an d
=
'
C
5
.
Bv
d
an
if A
=
aries as
h
0, t
A
en
B
+
an
VA B
d
will
.
6
.
If A
7
.
P
v
arie s as
B 0, t
direc tly
aries
h Bv
en
Q
as
aries
v
in
d in vers ely
an
;
e rs ely as
h
§
w
P=
als o
R ;
as
en
—
/48 d
h
P
1
3
i
t
t
v
h
v
If v i
+
8
y
y p
y
h
m f tw qu i i f wh i h v i
If y v i
9
d if y
d i ly
h i v ly
wh
d h
d
h
i
wh
d
u
b
t
w
d
fi
q
y
y
h
l
m f w qu i i
If y i
u
f wh i h v i
10
q
i v
ly
d i ly d h th
d if y
9 wh
fi dy i
mf
ly
If A v i
di
h
ly
t fB
d i v
11
qu
th
ub f 0 d if A wh B 6 d C fi d B wh A 2
9
dR
an
4
ar es as
x
.
rec t
as x an
1
—
en x =
3 ,;
s e
.
re c t
as x
3;
n
,
t
t er
n
to t
a
an
s o
x
e o
e c
su
x
2
o
ar es as x 2
at o n
e su
o
an
ers e
n
6
=
t t
en x =
4,
an
.
es
on e o
=
an
;
ar es
one
c
een x an
o
?
es , o
an
;
e
t
t t
an
as x
e
2
2
o
e rs e
n
an
ar es
c
1
en x =
an
n
2,
or
.
re c t
t
as
3
=
an
,
e
x
a
er as x
e o
ar es
.
an
t e
t e o t er
n
t e
3;
=
en
ro
,
ar es as
.
=
Qw
fin d
are ro o
e s
=
en
25
o
=
an
2,
as
ers e
4
=
en
n
1
d 0
2
h
t
i
y
2
fi d th
b w
i
v
d
d
h
wh
l i
d
t
d
p
y
d C
If A v i
B d C j i tly wh il B v i
D
13
v i i v ly A h w h t A v i D
If y v i
h
m f h q u i i f wh i h h fi i
14
h
d if y
dv i
hi d
wh
d h
fi
d
wh
wh
wh
d
7
y
y
y
i g
di
f mh
b d y f ll f m
i
15 Wh
i
l
h
l
l
i
f
b
d
f
l
v i
u
h
i
m
h
b
f
f
i
g
p i
y
q
h ugh 5f i
d m
d it f ll i
d?
h w f
i f ll i th
Al h w f d
d?
v th
Gi
12
.
en
e re at o n
n
=
1
n
ers e
as
e en
aries as 2
x an
+
an
ro
2,
c o n s tan
t, t
= 1
1,
dt
a
var es as z
x
at z =
e
ar es as
4 02
ro
o
ar
a
t
t
2,
2
x =
en
t
are o
n
t
5
a
t
e
se c o n
n
4
re e
3
an
re s t
t
s,
e
e
t
10
e
o
r
3;
ts
t
as
ar
s ec o n
2
as
,
’
an
.
an
en x =
ro
s
,
t
ar es
e
ar es as
o
=
a
e s
o es
a
e su
an
o
ee t
t
e
o n
ar e s as x , an
en x =
en
.
t
e s e co n
'
an
s
,
ar es as
.
so
et
ar es as
ar e s
o n
v
+y
.
.
t
at x
t t
es o
2
as x
en x =
s tan c e
o es
t
ro
n
a
a
:
n
rs t
e
0
=
an
n
e en
t
c
s
en
.
e
a
10
s tart n
o
s eco n
a
s
s
C H A PTE R I V
.
A R I TH M E TI C A L PR O
N N Q u ti i
wh h y i
h f h f ll w i g
D E FI
Pro gres s io n
38
.
f
eren c e
.
u
Th
s
I TI O
eac
Pro gres s io n
t
an
.
en
t
e
o
t
e
o
s a id
e s a re
to b
in A rit h
e
f m
s e rie s
n
.
de c re a s e b y
o r
n c re a s e
o
G R E SS I O N
or
s
a
co
mt i
mm
dif1
on
A rit h
an
c al
e
mti
e
cal
3, 7 , 1 1 , 1 5
8 , 2,
a,
-
d,
+
a
4 , — 10
2 d,
+
a
mm
f m h wh h
mm f
a
3 d,
+
fu
u
m
fe re n c e is o n d b y s b t ra c t in g a n y t e r
o n dif
of
Th e c o
ic fo llo ws it
ro
t at
I n t h e rs t o f t h e ab o e
th e s e rie
o n di fe re n c e is 4 ; in t h e s e c o n d it is
h
i
n
l
t
e co
e
ex a
p
t h e t ird it is d
s
m s
h
39
we
n o
t ic e t
th an th e
hu
T
n u
s
h
ex a
mi
a,
at
mb
in
f
er o
la
3
y
d, ge n e ra ll
n
th
,
To
Pro gress io n
hv
fin d
40
.
L et
th e
n
th e
b e th e n
we a e
m
t er
,
p
um
d,
ih
“
th
e en
e s er es
n
er
s a
er
s a +
er
s a
t er
s a
mi
l
su
2 d,
a +
e co e
er o
th e
s
mth fii i t f d i
mi th i
t mi
2d
5d ;
t mi
t mi
1 9d ;
u mb
:
m
f t er
(
a
mof
a
s,
n
nu
o
les s b y
s
one
.
l) d
(p
.
d if l de n o t e t h e la s t ,
1) d
an
o r
.
mb
er
of
mi
ter
s
n
A rith
mti
e
ca l
.
de n o t e t h e
b e r o f t er
a
s e rie
t
r
e
y
th e ter
an
th e
n
+
a
20
If
th e
n e
6
an
v
.
I f we
.
fi
.
fi
ms
.
rs t
m d th
te r
A ls o
,
e co
mm
on
le t I de n o t e t h e
f
di fe re n c e ,
la s t t e r ,
m
d
an d
an
n
s
G
A R I TH M E TI OA L
th e
re
i
u
q
re d s u
m th
PR O R E S S I ON
29
.
en
l
(
an
d, b y
w iti
r
s =
n
h
t
e
g
s e ries
h
re
— 2d +
l+ (l
A ddin g t o ge t
in t h e
)
er
t
h
es e
t wo
v
+
e rs e o rde r,
(
a +
+ a
s e rie s ,
to
l= a +
(
n
n
m
ter
s
1) d
-
(
2
a
{
.
1 ) d}
n
h v h u fu f mu
h
h
m
u w u y wh
h
h s
w
u u v v u
u
fi
m y
h f mu
y u
h
mh
f mu
w f
fu
v m
u
m
h h
u f
u
f mu
m
m
F d
m
5
mm d
m
H
1 h
a e t re e
I n t h e la s t a rtic le w e
se
l or
lae
in e a c
o f t es e a n
o n e o f t h e le tt e rs
a
d
n o te
e
y
y
a n t it
n kn o
n
a re k n o
n
F
the
e n t h e t re e o t e r
or
q
in s t a n c e , in ( 1 ) if we s b s tit t e gi e n al es fo r s , n , l, we o b ta in
a t io n
f
o r
n din
a
an d si
i
l
a rl
or
l
ae
an e
i
n the o t er
;
q
g
e c an ic a l u s e o f t e se
B u t it is n ec e s s a r t o g a rd a ga in s t a t o o
l
l
il
l
o te n b e
n d b e t te r t o s o l e s i
e
ae, a n d it
o
e n e ra l o r
p
g
a n a c t al re e re n c e t o t h e
s t io n s b
a
n t a l ra t e r t a n b
e
e
q
y
y
re
i
s it e o r
l
a
q
41
.
.
.
.
E x a p le 1
e re th e c o
th e
su
in
.
on
th e
su
o
f th e
iffe ren ce is 1
m
—
17
2
s e rie s
;
en ce
{
l
2
5
t o 1 7 te r
8,
,
s.
fro
x
17
—
—
2
17
x
31
2
_
m
E x a p le 2
4 00 : fi n d th e n
I f n b e th e
.
n
2635
.
mf
i
i 5 th
m d th mm d iff
m th f m( 1 )
Th e fi rs t t er
b er o f ter s ,
um
um
b er
o
f te r
s,
o
a
e co
an
en
4 00 4
wh
e n ce
s er e s
it
ro
}
2
16
.
s
,
o n
e
las t 4 5,
e re n ce
.
an
d th e
su
m
30
HI
If d b e th e
mm
iffe ren c e
th e 1 6
45
wh
AL
d
on
co
GH E R
m5
‘h
m
.
1 5d ;
te r
2s
d
en c e
G E BR A
.
hm
b
e
rit
e t ic a l Pro gre s s io n
A
y
l
el
d
e t e r in e d ; fo r t h e da t a
t
e
b
i
n ,
s e rie s c an
e co
e
p
g
will
lt an e o s e q at io n s , t h e s o l t io n o f w ic
rn is h two s i
f
o n dif
e ren c e
h
rs t t e r
an d t h e c o
v
t
e
i
e
g
“
um
h
Th e 54 an d 4 ter s o f an A P are — 6 1 an d 6 4 fi n d th e
E xa
le
fu
42
v
If
th e
.
d
23
ter
ter
mu
s
u
m
fi
m
p
m
two
an
.
m
If a b e th e fi rs t ter
m
h
w
an
6 4 = th e
d
we o b tain
en ce
d th e 23
ter
N
N
.
;
.
eren ce,
er
= a
er
a
g
7 1g
a
16A
22d
a
.
h
u
W h en t re e q an t itie s
D E F I I TI O
iddle o n e is s a id t o b e t h e
Pro gre ss io n t h e
t h e o t e r t wo
43
.
h
Th
44
m
h h
.
on
4
a
t
m
rd
u
mm diff
t m + 53d ;
= + 3d ;
t m
co
6 1 = th e
an
m
y
mm
m
d d th e
an
,
an
u
.
.
f
o
.
us
is t h e
a
fi
To
.
nd
a rit
th e
h mti m
e
h
a ri t
c
mti m
e
h
.
e en
d
a
e in
g
e
u
q
a
l to th e
co
mm
ean o
f
en c e
v
d
d
a
.
f
b
.
hm i m
et c
e an
.
m
di fe ren c e
on
a
A —
z
an
a rit
mu h v
.
b —A
wh
c al
b etween two given qu an ti ties
ea n
c
w
b et
e an
u
hb
e
.
L e t a an d b b e t h e t wo q a n t it ie s ; A t h e
st
a e
T e n s in c e a , A , b a re in A P w e
e ac
h mti
m m
in A rit
arit h
et ic
a re
g
u
m uh h
y
B e t we e n t wo gi e n q an t it ie s it is alwa s p o s s ib le t o
in se rt an y n
b er o f te r s s c
t at t h e
o le s e rie s t h s
o r e d s a ll b e in A P ; an d b y an e x t e n s io n o f t h e de n it io n in
s in s e rt e d a re c alle d t h e a ri th
A rt 4 3, t h e t e r s t
ean s
etic
45
um
.
f m h
m hu
.
.
m
E x a p le
to
ud
an
hm m b w
m
u
m
mw
m
P
wh h
mm d
etic
.
h
wh
t
m m
I n c l in g th e ex tre e s , th e n
fi n d a s e rie s o f 22 ter s in A
L et d b e th e
co
on
.
,
en c e
d= 3,
d th e
re q
an
d th e
u dm
ire
ean s
b er
of
o
et
een
4
s erie s
e an s are
“d
m4
ter
is 4 , 7 , 1 0,
7 , 1 0, 1 3 ,
an
d 67
.
.
h
hv
f t er s ill b e 2 2 ; s o t at we a
ic 4 is th e fi rs t an d 6 7 th e las t
.
ifferen c e
6 7 = th e 2 2
en
u
fi
.
arit
In s ert 2 0
.
wh
21 d
6 1 , 6 4, 6 7
58 , 7 1 , 6 4
.
e
G
A R I TH M E TI C A L PR O R E SS I ON
To in s ert a given
46
two given qu a n tities
n u
.
mb
of
er
ith
ar
31
.
mti m
e
c
ea n s
b etween
.
Le t
a an
v
d b b e t h e gi
u
h
fi
qu
en
m
I n c l din g t h e e x t re e s t h e
s o t a t w e h a e t o fi n d a s e rie s
rs t , a n d b is t h e las t
a is t h e
v
h
w
co
mm
an
b
(
2)
a
(
n
1) d ;
b
a
n
1
re
u
i
q
re d
m
h u
um
l
’
m
.
m
n
.
+
2;
ic
wh h
.
an
h
d th e t
n +
umb
in A P is 2 7 ,
ers
ree n
.
co
1
.
.
mm
d
d th e
an
h
ifferen ce ; t
on
en
su
mf
o
h
re e
wh
o se
th e t
.
um
b ers
re e n
are
9
4
5;
d, 9 , 9 + d
.
(9
d:
en c e
d th e
“‘
mw
.
—a
b
(
)
n
b er, d th e
a
9,
h
hm
um
H en ce
an
e an s
te r
1
u +
L e t a b e th e
iddle n
b ers are a d, a, a + d
wh
m
2 ( b —a )
E x a p le 1
Th e s u
o f t
t e ir s q ares is 2 9 3 ; fi n d t e
e n ce a =
m
e an s a re
u +
wh
f
m
b —a
n
er o
m
‘h
n
th e
en c e
d the
n
u mb
f te r s
ill b e n
2 t e r s in A P , o f
er o
f
o
n
difi e re ii c e
on
en
h
u mb
n
th e
n
.
L et d b e t h e
t
t ities ,
an
n
um
m
m
b ers
are
E x a p le 2
Fin
te r
is 3 n
1
.
4, 9, 14
d
th e
an
dn
.
su
m f th
o
m
fi rs t 1)
e
te r
s
.
u
By p ttin g n
:
1,
:
v
re s p ec ti ely ,
p
m 2 l t t m 3p
f
m § ( 2 + 3p 1 3
fi rs t ter
su
:
=
er
as
,
9
.
3
4
3
,
s
er
o
7
2
,
m
t 17 t m
t 19 t m
t o 2 0 te r
4 9 , 4 4,
o
=
=
E XA M PLE S
2, 3i,
we o b tain
er
.
s
.
s.
IV
.
a
.
1 ;
o
f
serie s
32
GH E R
HI
4
Su
5
.
Su
6
.
Su
S
7
9
u
,
,
7
Su
.
to
m5
3
m 3b
m2 b
+b
m2
s
.
m
t 24 t m
t 10 t m
t 50 t m
t o 1 6 te r
—7
1
/3
s
.
s
er
o
.
.
er
o
.
s
er
o
,
.
m
t er
n
,
Su
8
m3 g
m3 7 5 3 5
m
m
m 3 3 J3
G E BR A
AL
s
.
.
m
t o 2 5 te r
,
s
.
N
10
.
Su
11
.
Su
12
Su
.
—
a
a
2 a — 5b , 3 a
,
3b, 6a
4a
,
to
t o 2 1 te r
2
hm i m
i hm i m
i hm i m
i hm i m
mf h fi
w
w
w
w
s
.
I n s ert 1 9
arit
et c
ean s
b et
een
14
.
I n s e rt 1 7
ar
t
et c
e an s
b et
een
I n s ert 1 8
ar
t
et
e an s
b et
e en
.
16
.
I n s ert
17
.
Fin
d
x ar
th e
c
et c
t
su
t
o
b et
ean s
rs t n
e
d
.
.
s
ee n
dd
o
mi
I n an A P th e fi rs t t e r
18
fi n d th e iffe ren c e
.
s
.
.
.
1
13
15
n
4
3%
x
an
u mb
e rs
.
d
an
3 5x
2
93
d
an
4 15
.
an
d 1
d 3x
.
.
.
m29
2 , t h e las t t er
th e
,
su
m1 55;
.
m
mf
m
o f 1 5 t er
Th e s u
19
rs t te r
e n c e is 5; fi n d t h e
s o
20 Th e t ir t e r
o f 1 7 ter
fi n d th e s u
an
.
fi
A P is 6 00,
an
.
.
an
d th e
co
mm
d iff
er
on
.
d h
h d mf A Pi
v h t mi
m
m
umb i A P i 7 d h i p du i
m f th
Th
21
fi d h m
u mb i A P i
d h
Th
mf h
mf h i
22
ub i
fi d h m
Fi d h
m f h i wh
mf
mi
23
P
Fi d h m f
m f h i wh p
mi 7
24
m fp m f h i wh
mi d b
25 Fi d h
Fi d h
mf
mfh i
26
o
.
504 ;
n
s
.
t
e
4 08 ;
n
n
s
1 8,
s
.
.
t
an
e se
en
t
er
s
30
.
ree n
e rs
re e n
e rs
n
.
.
s
2
s
1 2,
t
an
,
e r
ct
ro
s
.
o
e su
.
es
o
e su
.
c
te r
n
s
m
3a — b
a
m
m
t o 4 0 t er
n
t
t
t
e su
e su
t
e
n
.
.
an
t
t
o
e su
e r
.
1 5 ter
o
3 5 ter
o
.
n
t
e su
o
.
n
t
e su
o
2a
n
2
s o
t
e s er es
t
s o
e s er e s
t er
s o
t
e s e r es
t er
s o
t
e s er es
1
6a
3
a
’
2
a
o se n
th
‘h
o se
o se n
te r
ib
s
te r
t er
4n + 1
.
+2
.
s
n
s
.
G R E SSI ON
A R ITH M ETI C A L PR O
33
.
m v
wh
I n a n A I it h III e tic a l Pro gI e s s io n
47
en 3 , a , d a
i
n ,
e
g
al e s o f n
we h a e t h e q u a dI a t ic e u a t iO I
de te r in e t h e
q
m
.
to
v
v
u
0
. a
2
wh
h
v
o
s it i e
p
(
l
- -
.
i
1)d
n
h
ffi u y
d in t e gral t e re is n o di c lt
in in te rp re tin g t h e re s lt c o rre s p o n din g t o e a c
I II s o e c a s e s
i
e n fo r a n e a t i e
a s itab le in te r re t at io n c a n b e
al e o f n
p
g
g
bot
en
ro o
a re
u
u
m
ts
k h
ta
.
an
a
h
v
my
mmy b
How
E x a p le
e n t at th e su
an
m
te r
e 66 ?
g
s
o
f th e
series
-
9,
.
m
v v u
.
mu t
—6
,
b
s
e
—l 8 + n
(
{
7n
1 1)
(n
n
m
k
I f we ta
th e
su
1 1 te r
e
m f wh i h i
c
o
s o
11
:
—9 , — 6 , — 3 ,
s
66
4
.
hv
we
serie s ,
f th e
or
a e
6 , 9 , 1 2 , 1 5, 1 8 , 2 1 ;
.
h m
u m
hu h u h
v u d
d
d
d v
m
w
u
d w h h wh h
If we b egin at th e la s t o f t e s e te r s an d co u n t b a c kwards fo r ter s , th e
s , alt o g
th e n egati e s o l tio n
irec tly
is also 6 6 ; an d t
o es n o t
su
ean in g,
an s er th e q es tio n p ro p o s e , we are en ab le to gi e it an in telligib le
ic
it t at to
a n d we s ee t at it an s e rs a q es tio n c lo s ely c o n n ec te
th e p o s iti e so l tio n ap plies
m
w
u
h
v
We
48
o llo
.
th e
f wi
Th e
u
n
e
u
q
.
c an
ju tify th is i
w
a
y
g
a tio n
n
s
t e rpre ta t io n in t h e ge n e ra l
.
mi
t o det e r
dn
2
n e n
+
is
2
a — d
)
(
u
u
n —
2s
z
0
Sin c e in t h e c a se n de r dis c s s io n t h e ro o t s
by n l
o pp o s it e s ign , le t u s de n o t e t e
e rie s c o rre s p o n din g t o n l is
te r
o f th e
m
s
s
a
m
hm
(
h
1) d ,
n
u
f t is e q
—n
an d
,
o
u
.
a t io n
hv
m
2
s
d we h a ll s
H
.
H
.
A
.
h
e
(
a + n
h
— l
,
w t h a t t is is
m
e
u
q
a
l to
s
.
)(
—
d)
a
e
Th e la s t
mm
a n d c o n t b ackwa rds , t h e c o
t
er
a t t h is
i
n
if we b etg
a
—
t
I n Is is
h
f
n
e
b
an
d
t
s
u
o
d
e
t
d
n
o
e
e
u st b e d
dide ren c e
,
y
an
in
case
on
34
GH E R
H I
Fo r t h e
p
ex
AL
2
u
(
2a
re s s io n
2 an 2
2n
1
s in c e
p
ro
u
d
n
ct o
s a t is
2
f th e
h
fi
m
.
E
a
xa
mu
o
n
m
pl
e
v lu
w
o
2n
l
n
28 )
h
u
f
e o
myt m
an
s o
er
(
f th e
n
2
hu
T
s
th e
2
d
2a
n
2
)
)
O,
an
d
n
l
n
wh
50
We
.
xa
m
pl
1
+1
e
f7n
um
a
m
dd
2
.
uh
h
u
s e rie s
26, 21,
4
n o
ex act n u
Th e
.
4n
hv
a
v
s
v
s
k
b e ta
m
w
d
co
mm
d
on
udh
h
u
m
s o
an
to
en
= 74 .
I t ill b e fo n
i s less t an 7 4
s c e lla n e o
um
Exa
s
t
mpl
es
ter
s
.
e tic s e rie s
are
ms
ce o
m
m
hm
h
ifferen
su
.
f two arit
o f t e ir l l tk ter
s o
th e
at
in th e
.
f th e two
s erie s
be
a
l
d1
,
an
d
.
e
N o w we h a
m
.
f n te r
+ 2 7 ; fi n d th e ratio
L et th e fi rs t ter
d2 res p ec ti ely
We
m
e
m
.
or
m Mi
so
th e
is
4 ) ( 571
-
b e r o f t er s is 4
ile th e s u
o f 8 te r
n
is gre a ter,
e
2 a1 +
(
2az +
(u
n
—l
)
1 ) d2
v u
to fi n d th e
al e o
dl
f
a
7n + 1
4n + 27
l 0d1
l
“2 +
h
0
1 0d2
en c e ,
u
b y p ttin g n
21,
:
we
b tain
2 a, —
l—2 0d1
t
2
l) d
57 n + 1 4 8 = 0
n =
o
(
2
n
t to 7 4 ?
591
0 2,
n
2s
n
o
2
s,
:
—5
E
ratio
1) d
d
2
n
d
( 2
d
2
.
f n is ra c tio n a l t e re is
c o rre s o n ds t o s c
a s o l t io n
p
a
h h
H
.
1
dn
2
a — d n
)
(
o f t is e
a t io n
q
ts
W e n th e
49
b e r o f t e r s w ic
n
2
es
ro o
n
4
8
(
f
§
—
l
G E BR A
hu
s
th e
u d
re q
m
ire
2 a2 + 2 0d2
ratio
E x a p le 2
I f S]
o s e fi rs t ter
s e rie s
fi n d th e
1 , 3 , 5,
wh
.
is 4
3
a re
s a re
1 , 2, 3,
al
f
v u
e o
3
111
’
.
SQ ,
m
,
4
148
th e
S 1 + S 2 + S3 +
an
uwhm
s
s
d
+ Sp
o
f
o se c o
.
m f
mm d iff
n
te r
on
s
o
hm
a rit
et ic
e ren c e s are
36
GH E R
HI
9
Su
.
mth
1
1
e s e ries
J
1 +
m
m
’
x
m
11
t
ht
s
“
“
g
,
s
4 9,
e
,
ter
er
su
o
s
n
A P is g,
an
s o
r
o
mt m
“!
er
16
mk
a
.
.
e
er
o
289 ,
s
s
v
c res p ec ti ely , s
b,
d th e
an
.
in A P is 2 4 ,
e ers
su
rt
o
th
mf
te r
o
h
er
an
mf
s
.
o
e
er
o
s
s
du
.
f th e
s eries
e r
.
m
i
t
m
g
d th e
an
.
h i p du t i
d u h th t h
d d hi d
ro
in A P , an s c
c t o f th e s eco n
are
t o th e pro
s
A P is 9 ,
an
hw
m f g t mi
d t
an
.
.
whic
arts
r
o
my t m
H ow
3 06 ?
,
.
.
fu p
fu hi
fi
Th e p
.
s
m f 1 7 t mi
a re a,
.
.
.
15
m
t er
n
.
v
u
d th e
f an A P
s o
er
o
D i ide 20 in to
14
rs t an d
t
h
e
c
t
o
f
r
o
d
p
in th e ratio o f 2 t o 3
.
an
to
’
Jx
x
m
“
h
r
o
e su
.
’
-
m fp t m f
m fp 9 t m
mff u i t g
Th
d th m
Th e s u
12
fi
n
h
d
t
e su
p;
13
9 4 5; fi n
1
1—.
.
(g
a
.
G E BR A
.
I f th e p
.
1
mi
I f th e s u
o f 7 t er
10
s
o f n te r
fi n d th e s u
.
AL
er
s
t
a
t
an
e
r
fi
n d th e
;
p
s
mu t b
9 , 1 2, 1 5
c
s
k
ta
e
to
en
m f A P i 2 + 3 fi d th t m
m f mt m f A P i t th m f t m
If th
18
mt M h w th t th m t mi t th t m 2m 1 i t 2 l
P v th t th
m f dd umb f t mi A P i q u l
19
idd l t mm
ultipli d b y th u mb f t m
t th m
l
fi
h
5
3
ll
u
f
d
t
t
m
f
20 I f
)
(
p
Th
umb f t m i A P i v ; th m f th dd
21
t m i 24
v t m3 0 d th l t t m d th fi t b y
f th
um
b
ft m
fi d th
tw
t
Th
f um
22
b
h i ti g f 3 t mi A P
d th
m f h t i 1 5 Th mm d iff
f th fi t
t
b y 1 th
th
mm diff
i g
t
f th
d
t
d th
t
t
f
t
h
t
i
t
h
d
t
u
fi
d t
7 t 8 fi d
p
p du t f th
th
u mb
23
F i d th l ti b tw
dy i
m
d t h t th
b tw
d 2y my b th
m th
m b tw 2 d y
17
I f th e
.
o
e
s
,
e
e
er
s
s
e su
ro
c
e n
o
e
o
m
s
o
o
er o
er
er
en
er o
e ac
rs
s
se
an
e
h
.
s
.
er
en
s as
71.
.
s e
.
a
.
ih
.
e su
er
as
o
.
ex cee
o
e o
s
e
er
s
rs
.
e rs eac
co n s s
e co
on
ro
e
on
c
o
n
o
ere n c e o
e ren c e o
n
e re a
a
on
e
b ein g in serted in
m
e
e en
e sa
eac
h
x
e s ec o n
e seco n
m
m
an
cas e
.
.
n
‘h
e as
se
n
e
se
,
rs
.
se
an
o
as
.
e
:
n
e r
o r er
e an
‘h
a
e
e r
een
e an
x an
,
.
I f th e s u
o f an A P is th e
t at I ts s u
fo r p + g t er s I s z ero
24
er
n
s
er
er
n
s
er
e
s e
o
as
n
.
.
an
.
o
er
er o
n,
e r
n
e su
er o
n
es o
n
s
s
,
s
e co
se
n
s
er
o se s o
e n
n ,
o
s
‘11
e n
va
n
.
.
an o
ora
n
s
.
‘h
.
e e n x an
ean s
er
.
an
s o
e
an
ers
.
e
e su
e e
rea er
s
a
ere are
.
an
e
e n
n
“
an
s o
er
er
o
,
ter
a
e n
.
n
o
s ==n
.
it
e
ro
.
o
o
e su
.
2
su
2
mf
sa
e
or
p
as
m
fo r 9 t er
s, s
hw
e
C H A PTE R
G E OM ETR I C A L
N
.
hu
T s
re s s io n
g
ea c h
.
o
f t
.
G
PR O R E S S I O
Q u an titie s
D E F I I TION
Pro gress io n wh e n t e
51
V
s a id
a re
h yi
h
f ll wi
o
.
t o b e in
Geo
mt i
e r cal
de c re a s e b y a co n s ta n t fa c to r
n
s e ries
o r s a Geo
e t ric a l Pro
g
n c re a s e o r
e
N
o
f m
.
m
3 , 6 , 1 2, 2 4 ,
1
1
3
g
a , a r, a r
f
3
2
ar
mm
h h mmdi
mm
Th e c o n s t an t a c t o r is als o c a lle d t h e co
i
oun d b
d
i
i
n
an
r
a t w ic
i
d
t
e
b
t
y
g
y
y
it
I n t h e rs t o f t h e a b o e e x a ples t h e c o
1
t h e s eco n d it is
; in t h e t ird it is r
f
v
fi
.
m
h
5
I f we
.
m
v
-
52
h
ex a
mi
n e
s
th e
hu
T
ar
,
m
m
ar
,
the 6
mi
mi
s ar
th
ter
s ar
t h e 20
y
d, ge n e ra ll ,
I f n b e th e
t e r , we a e
an
m
hv
N
th e p
n
u mb
er o
N
D E F I I TI O
Pro gres s io n t h e iddle
t h e o t h e r t wo
53
.
.
m
.
W
th
mi
mi
m
ter
te r
f te r
t
h
on e
en
is
f
o
a lwa
r
s
y
les s b y
on e
.
2
ter
th
on
y
,
ra
th e 3
s
e
“
we n o t ic e t a t i n a n y ter
th e i n dex
tha n th e n u
b er of th e ter i n th e s eri es
m
d it is
at e l
ecedes
r
p
ra t io is 2
in
s
3
’z
h
r a t io , a n
.
e rie
a , a r, a r
on
s ar
f’ “
s ar
s , an
d if l de n o t e t h e la s t ,
u
s
t h re e q a n t it ie a re in
et ric
c a lle d t h e ge o
m
G eo
m
ean
o r n
mt
e ric a l
bet
w
een
38
HI
To fi n d th e geo
GH E R
mt i m
L e t a a n d b b e t h e t wo q
T e n s in c e a , G, 6 a re in G R ,
h
u
an
b e in g e q
u
a
l to t h e
b
0
G
a
mm
co
.
mt
e ric
m
e an
.
ra t io
on
ab
G
h
—
To i n
.
two given qu
L et
In
t
fn
Let
an
sert
ti ti es
h
e re
b
i
v
en
g
e
th e
e
mb
n u
er
.
f
o
eo
g
mt
i
e r c
m
ea n s
b etween
.
w ill b
v
s
co
an
n
+
n
t it ie s ,
m
wh
.
.
th e
n
t
so
a
h
n
u mb
er o
we h a
at
is t h e
fi
v
rs t a n
e
f
m
e an s
to fi n d
.
a
d b t h e la s t
.
rat io
on
b
en
u
q
en
2 te r s ;
G P, of
ic h
e
mi
mm
2 te r
r
a
d b b e t h e gi
a an
a ll
se rie s o
t
b
J
G
’
en c e
54
h
ti ti es
an
t it ie s ; G t h e ge o
2
t
.
.
eac h
w
G E BR A
b etween two given qu
ea n
e r c
AL
th e
(
2)
n
‘h
m
ter
ar
1
( )
H
th e
en ce
v lu f u
o
e
a
n
m
E x a p le
hv
We
h
s ix t
re
i
u
q
.
a e
r
d
m
a re
ean s
a r,
mt i m
mi G P
I n s ert 4 geo
e r c
to fi n d 6 ter
b e th e
co
mm
o n
s
n
rat io
h
t
e an s
.
.
o
;
en
5= th e
1
32
1
en c e
an
d th e
m
e an s are
Q
8 0, 4 0, 2 0, 1 0
.
w
wh h
wh
e re
r
1 60
f
ic
1 6 0 is th e fi rs t,
m
t er
an
h a s th e
d5
ee n
h
1 60r
h
,
b et
s ix t
5
w
ar
d in
.
L et
re
"
.
an
d 5 th e
G E OM ETR ICA L
.
Pro gres s i o n
Le t
a
m
te r
fi
To
55
s , an
n
d th e
mb
g
e
m
by
s
m u
mm
er
ub t
h
.
v y t mb y
er
hv
we
r,
,
(
r
—
"
s = ar — a
l)
s = a
Ch a n gin g t h e
si
ns
g
in
n
ud
N OTE
I t will b e fo n
fo r s , s in g ( 2 ) in all c as e s
a
.
E
c
xa
Th e
th e
1
mwhi h i
fo r
su
m
pl
e
co
m
.
Geo
n
mt i
e r ca l
:
um
e ra t o r a n
Su
mtim
e
mth
o n ratio
es
v
en
e x c ep t
mul
s so
mm
co n
l, th e fo r
a
u
ra t io
th e
n
,
n u
mb
er o
f
l)
1
1
( )
d de n o
l
(
r
1
r
ien t to
wh
en r
my b
n a t o r,
)
mmb
h
bot
er
e
T’
2
w
ritten
.
1
s eful .
m
2
t o 7 ter
e s e ri e s
h
en c e
m giv
fo r
s
mul
b y fo r
a
2
( )
s.
ab o
en
is p o s itive a n d grea ter th a n 1
e
r
mi
"
re
a
1
;
l) ;
(
"
a r
(
a
an
s
ar
r
r
Sin c e
mi
"
ar
ra c t io n
o n
" 2
ar
rs
t-
ter
e
a
2
ar
rs
u
of
er
n u
a
39
.
.
f
n
en c e
o
b e t h e irs t t e r , r t h e c o
re
re d
d s th e s u
i
n
T
e
q
mu ltiply i
H
mf
su
G
PR O R E S S I O N
.
v
e
40
HI
56
Th e
su
mt
o
F mt h i
s
ro
a
su
,
as
11 —1
a
ro
57
u lt
1
v my
wy
h
h
v
ufi
l
as e
e
p
e
as
be
c an
n e x t a rtic le a
F mA
ro
.
.
s
.
I n th e
GE BR A
rt
hu
T
.
md
a
m
s
b y t ak in g
f
t o di fe r b y
e
e n e ra l c a se
o re
v
is
r
mll
p
a
ro
p
f
er
ra c t io n
; t
h
s
is t h e
er
a
e re o re
a
n
a
n
e
”
o
r
c en
s
ea e
e
mb
or
f
er o
m
th e
du
D
umb
n
s
y
m
s
m of
ter
E xa p le 1
c t is 2 1 6
p ro
en o
s
b rie fl
o re
er
u
n
as
t
n
we
u mb
l
s
p
er
ea e
s s ed
.
,
t
t h e gre at e r t h e
we
ar e,
by
ro
°
u
q
en
y
tl
f
W
a
e
a
n
an
1
th e
mk
mll qu
c an
as s
o
a
u
v al e o
r
su
an
f
d
mf
o
tit
y
as
.
re
s
nu
e s e r es
o
s
er
c
litt le
as
l —r
c o n se
an
a
n
u ffi ie
f
ar
en
v lu f
d
th
f b y mki g u ffi i ly l g
i
t m f th
diff f m
w pl s
Th i
su lt i u u lly t t d th u
th e
n
.
e 3
1 —r
u
a s
is dis c
g
55 we h a
.
m
y
a
S pp o s e
m
an
ter s b e
it app e ars t a t h o we e r
o f t h e a b o e se ries is al a s le s s t an 2
A ls o we
l
a ke t h e
rac t io n
a in
s f c ien t l
a rge , w e c a n
n
g
re s
e su
s
er
o
s
AL
m
te r
n
m
t h t b y mk
1
w
s
ll
m
2
m
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f m2
t ak e n t h e
see
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.
G H ER
,
th e
Fin
.
su
t
s :
decrea sin g Geo
a
d h
s a e
a
mt
in
o
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fi
n
umb
mt i
e r ca
su
mof
an
fi
in
l Pro gress io n is
n
i te
a
1
r
a
i ty i s
e rs
th e
l —r
in G P
.
.
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wh
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mi
s
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.
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h
t
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en
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h
en c e a
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an
d
GE OM E TR IC A L
G
PR O R E S S I O N
41
.
6
wh
2
3
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hu
T
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m
m
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m
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42
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10
11
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C H A PTE R
H A R M ON I C A L PR O
G R E SS I O N
61
D E FI
.
N
o
mm l P
H ar
A
n
y
u mb
wh
n
N
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re e
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t
a
b
C
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6
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Pro gre s s io n
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m
G R E S S I ON S
h qu
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TH E O R E M S C O
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PR O
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W IT H
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to
e
b e in
m
te r
s
to
b e in
m
H
ar
a re
in
o n
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H
ar
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62
Th e
a re i n A ri th
rec i ro ca ls o
p
mti
.
B y de
fi
n
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f qu a n ti ties
ca
l Pro gres si o n
itio n , if a , b ,
c a re
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in H
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H a r o n ic a l p ro p e rt ie s a re c ie
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T e re is n o ge n e ra l o r
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i
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Q
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s an d
H P a re ge n e ra ll
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A
p p
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m
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mk
48
HI
64
GH E R
G
A L E BR A
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To fi n d th e h a r
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1
u
b e t h e t wo q
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P ,
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d
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I n s ert 4 0
mm
mf
h
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l
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F mt h
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A
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le t d b e th e
13
d
an
,
hm
v
b e t h e a rit
e t ic ,
eo
g
a n d b , we h a v e
ro e d
p
AH
o re
o se
7
h
d t
ere fo re
th e h ar
41
a
b
6
g5
36
mt i m
e r c
su lts w
e an
e see
—G
mt
e ric , a n
6
a
Th e re
wh
ee n
7
35
,
a
.
41d ;
I f A , G, H
e t we e n
w
b et
ean s
an
1
hm m
ari t
m
en
6=
T
b
a
’
ab
G
b etween A
t h at
2
an
dH
.
mi
d h ar
on c
H A R MON ICA L
h hi
p
ea n
an
w ic
s
m
m
f
o
ea n
if a
o s it i ve
d b
an
G R E SSI ON
v
s it i
o
p
a re
o s it ive qu
t
w
o
y
p
PR O
t
e
49
.
h f
th e
e re o re
tit ies i s grea ter th a n th eir geo
an
mti
mt i
a ri th
e
c
e r c
.
f m
v u
h f
u
w
h
A H , we s e e t a t G is in t e r
ro
t h e e q a t io n G
A ls o
e e n A a n d 11 ; a n d it h a s b e e n
al e b e t
ro
e dia t e in
e
d
t
at
p
’
11 ; t a t is , th e a ri th etic, geo etri c , a n d
0 , t e re o re G
A
h a r o n ic e a n s b e tween a n y two p o s itive q u a n tities a re i n des cen d in g
a gn i tu de
o rder of
m
m
m
m
2
m
h
v
m
h
.
u u
u
fi
M is c e lla n eo s q e s t io n s in t h e Pro gre ss io n s a ffo rd s c o p e
66
fo r s k ill a n d in ge n it y , t h e s o l tio n b e in g o te n n e a t ly e ffe c t e d
Th e s t de n t
ill fi n d t h e o llo in g
b y s o e s pe c ial a rti c e
in ts s e l
.
m
h
.
sa
2
th e
wi h
3
sa
m qu
e
e
.
o rtio n
,
Co n
in
re
p
f
an
m
s
t it y , t h e
mm
.
at
f
o
re s
f
u
mw f m
f m ll
wi h
mu ltipli d
m will f m
di ide d b y
A P, but
ul i
t
n
.
r
t
e
g
A rt
[
.
v
y
e rs e l
m
,
a re
If
.
in G P , t
b
c
b
c
d
fq
o
e
x , xr , x r
b
'
2
C
,
2
dd
in g a b + a c + b e t o
dvd
(
a
+ b)
i i in g e ac
h
mb y
ter
o r
o r
an
o r
e
or
a
v
.
v
.
di ide d b y t h e
G P
it
th e
.
w h
.
.
hy
e
u
an
t it ie s in
,
in A P
are
.
.
e ac
h
s
,
a re a ls o
in
u
ed
t in
co n
hw h
mw
ter
t
e
e s ee
,
‘
c
(
( b + c)
(
a
+ b ) (b +
1
b +c
c
at
c
+
c
2
+
+
a
1
a
’
h
t
co n
tin
p
ro
o rt io n
b + c,
c
d p ro
u e
p
+ a,
a
ca
) (
at
+ cb +
b)
a+
a
+b
a re
mA
.
P
H P
.
a b are
are
)
1
’
t
.
.
a
a
,
a
s
.
.
it io n
n
a s e rie
ted b y
.
fi
f
s
e
mu ltipli d
m w ill f m
.
.
,
.
.
.
u
I f a , b , c,
s in c e , b y de
re s e n
is
th a t is
f w
A P be
an
m
b
h
u
diffe re n c e
on
al
E x a p le 1
H P
By
t
y
l th e te r s o f a G P b e
s
a n t it , t h e re s ltin g t e r
q
A
rt
o n ra t io a s b e o re
co
[
If
.
e
be
u
.
y
m u
m mm
4
p
m
th e te r
a ll
a n ew c o
t
sa
If
sa
w
.
m
m mm
.
f
I f t h e s a e q a n t it b e a dde d t o , o r s b t ra c t e d ro
ill o r a n A P
s o f a n A P , t h e re s lt in g t e r s
o n di fe re n c e a s b e o re
rt
e co
A
[
t h e ter
the
u
.
u fu
1
u
.
in A P ;
.
in A P
.
.
.
my
+b
a
are
50
o
HI
m
m
GH ER
G
A L E BR A
md
If t th e las t te r
E x a p le 2
f n ter s o f an A P b e c o n n ec t e
.
.
.
mm
th e c o
b y th e e q
d
,
d = 2a
v
Sin c e th e gi
u
is tr
en relatio n
fo r
e
H
by
en c e
s
u
u
(d
2a)
2
m
h
th e
uu
s
a
a
e ac
h h
o
ft
—
+ (q 1 ) d
th e
n u
fi
rs t n
—
q, q
—
— r,
1) d }
s
er
a
d by
mil
si
mp
f t er
o
s,
u
t
su
y
a rl
h
1
:
t
h
en
ter
f an A P
s o
.
.
are
in G P ,
s
.
.
hw h
t
e
at
a
r—1
(
+
)
r—
(
a + (s
a
d
+
1) d
[A rt
1) d
-
66
.
.
+ (q — l ) d
a
l
1
d
U
) }
{M
l
a
—
+ ( q l ) dl
{
—l d
V )
l
+
re e rre d
to
l
{M
-
mb
e
an
a
(
(
n
M
(
T-
l) dl
s
1)
-
3
6
in G P
.
.
o
a re
e
o
f
ten
s e rie
si
f
s n
,
an
d th e
as
su
th e
mf
o
-
su
m f th
o
den o t e d b y
3
1
—
n
i
n to
i
n
g g
n
G
— n
(
z
n
e
c
n
9il
S
an
b er
e s qu a res
f
o
th e
fi
rs t 11
.
L et th e
We
um
.
m f th
i
n d th e
f
.
hv
at
2 a) 2 ,
—
1) d
+ (q
{
dl
r — s are
m
t m
si
To
68
b ers
m
t
3.
a
um
.
n
m
“
b e rs 1 , 2 , 3 ,
Th e n
th
ter
b ers ; t h e n
a tu ra l n u
67
n
p
e
es e ratio s
{ + (p
—
a
{ + (q l )
en ce
m
v h
2
=
d
8 ds
( + 2 l) , p ro
atio n
su
a e
+ (p — 1 ) d
a
H
d s th e
hv
tatio n we
al n o
an
.
.
W it
3
.
ifferen c e ,
O;
d = 2a
I f th e p
E x a p le 3
q
— r r — s are in G P
,
q, q
y
(d
8a a
b s tit tio n ,
u
d
on
.
an
l:
a
z
.
2
2
2
3
n
— 1
n
1,
(
S th
)
3
’
=
3 n — 3n + 1 ;
2
3 (n
(
n
(
n
3 (n
3 — 2
3
2
3
1 —0
3
3
z
3
.
en
1
2
—3
(
3 (n
n
1)
1 ;
2)
1
n a tu ra
l
52
HI
f
v
ym w h
G H ER
AL
G E BR A
u
wh
hm
.
ju
v
I n re e rrin g t o t h e re s lt s we h a v e
st
ro
i
e
d
t will
p
ic h t h e s t de n t wi ll fre
b e c o n e n ie n t t o in t ro d c e a n o t a t io n
at ic s
i
t
i
n H i
e r M at e
W
ee t
e s a ll de n o te t h e
en tl
g
q
s e rie s
70
.
u
u
h
u
h
.
En
b y En ;
3
wh 2
wh i h t h
e re
c
b y 2n ;
m
f
fi
si n i
l
ac
e d b e o re a t e r
g
p
is t h e ge n e ral t p e
a t ter
m
m
E x a p le 1
mth
Su
.
y
Th e u
ilar fo r we s
si
n
b ers , an d th e
e s erie s
m
um
m
h h v w um
h h u
all
t
o
th e
t
o ft
er
su
mE
n
(
n
l
o co
a e
2
u
are s
w
m
L e t th e
su
Su
.
mb
e
S
mt
d
en o
2 2
:
te
n
"
2
82
1
h
h
8n
n
2
t
3
(
wh
mi
o se n
ter
+ 1)
( 271 4 1 )
s
2
71 —1
en
6 2 72 2
2
1
+ )
n
6n
(
n
4
-
6
2n
— 2n
E XA M PL E S
Fin
d
th e fo
( 2)
3
( )
ter
n
ea c
h
VI
.
o
.
a
.
f th e fo llo
wi g
s erIe s
n
2, 25
,
2 , 2 2, 3 3
h mi m
f u h m
m
I n s ert two
I n s e rt
u h mi
rt
o
r
ar
ar
on
m
u
+ 1)
(u
n
e s erie s
s
2—1
-
dw
2
by S
]
‘
f
.
+ 1 ) ( 2n + 1 )
m th
d
s o
Zn
t er
o n
m
l te r
al
ter
in a
g o n e ac
c o n s is tin g o f th e fi rs t n n at ral
6
E x a p le 2
o
s.
ritin
d by
n s, o n e
eir s q
mf
m
te r
n
an
su
.
1
‘h
th e
es
c
o n ic
ea n s
ea n s
b et
w
w
b et
ee n
5 an d 1 1
2
ee n
3
an
d
.
2
13
8n 3
6n
2
.
E X A M PL E S O N
G
TH E
PR O R E SS I ON S
m
I f 1 2 an d 9 2 are t h e geo et ric
4
ively , b e t ee n two n u b ers , fi n d t e
I f th e h an
5
m
mt
f 4 to 9
6
non c
b e in H
c
a
7
eq
I f th e
.
u l mp v h
to
a
,
t
e
‘“
t
e
.
tn
q
— r bc +
(q
)
-
N
l
(
on c
d
.
16
G P,
.
su
mf
O
p)
.
11
14
If t h e
an d
n,
,
(
m
.
.
d iff
e ren c e
r
b
3
.
n
.
3
+
,
an
d th e
m
ul m
—
e
7,
v ly
a,
e
,
p
ro
v
e
.
een
d
a an
c,
r
o
p
v h
t
e
at
c
a
wh
g
mi
s
.
th e A
n
12
.
— 2”
n
P,
.
o se n
mi
"t e r
s
n
.
15
—
.
m
4n 3
.
t e r s O f an A P are in
on
t at th e rat io O f th e c o
2
d
hw h
e
.
t
P is
.
.
mm
-
.
7b
17
m
th
u
m
b
i G
m
P wh
m d t m
m+ l t 1
If th
mf t m f
i
I f I,
.
f an A
differe n c e
o
.
18
.
an d
19
,
n
o se
.
o
th e
n
d
n at
u
th e
n
re o
er
n
f th e
mf
o
m
P , p ro
.
in H
w
hw h
a
s er e s
be
.
v h
t
e
m
mm
th e fi rs t te r
t o th e c o
on
at
P is
.
s Of th e s e rie s
2
+ 1)
u
an
(n
m
a
+ b n + cn
(6n
2
wh
o se n
.
ro
e
t
a
t
s
e
t
exac t
n
.
at
t
a
rs
o
h
tities t ere b e in s e rt e
e an s G , 0 2 ; a n d t wo
1
m
fi
ar
e o
e su
c an n o
t
e
e c
t
et c
t
t
fi n d th e
t er
n
e
et
e su
een
t
es o
o
een
et
t
d w i hm i
h m m
t
o ar
ar
een
e sa
e
an
o
p
s
.
.
et c
t
ea n s
o n ic
t wo
t wo
an
te r s
t e te r
e
s
g
e an s
et
ea n s
on c
o
s
ar
n
,
th
mi
th
fi t f i hm i m b w
h mi m b w h m
d
h t h v lu f q
li b w
p v
p
Fi d h
m f h ub f h m f A
22
h ii
ly div i ib l b y h
mf h m
I f p b e th e
21
an d
t
h
e fi rs t o f n
q
2
.
m
t er
n
If b e t ee n an y two q
ea n s A 1 , A 2
t wo ge o e tric
H 2 ; s e t at G1 0 ,
.
.
s are
er
s o
s e ries
4n
20
th
n
.
O
su
e rs
an
,
as
Fi
.
re e n
are
e su
m
te r
t
mb
ter
n
to
a
u
'
n
an
H
t o th e fi rs t te r
n
b , 0 re s p e c t i
= o
a
b
q)
.
w
— c
+
n
in
a re
P be
.
+ (p
b et
l to
s eq
s Of t h e s e ries
2
a
‘h
1
m
m
n
ca
ea n
— a
te r
n
t
10
13
th e
b
H
s Of a
r—
u
er
1
Fin
e
m
) t mi
m
~
eq
.
h mi m
ar
h
tities is to t e ir geo
an t i t ie s are in
th e ra tio
an
at
H P be
a
(
e an s , re s p e c t
on c
—a
te r
r
,
I f b is th e
.
th e
at
“
I f th e
8
9
ro
o
er
ar
o
z a
m t mf
‘“
ee n
e
s
,
t
t at t
e
ro
.
.
an
.
e an
.
If a , b ,
.
h mi m
d
hm
b tw
w qu
h h qu
p v
P h w h
1 2 to 1 3 ,
e an s as
e ri c
o
m
i m
w
.
53
.
n
er
s
n u
mb
an
d
e is
,
.
P,
.
umb
s
hw
e
54
HI
PI L E S
71
r
py a
mid
.
To
fi
on
a s qu a re
u
n
n
d th e
h
GH E R
SH
OF
G
A L E BR A
OT
mb
SH
AND
f
er
n u
sh o
o
.
t
E L LS
a rra n
.
ged i n
a
um
to p
h
b as e
.
h
h
h
w
y
h
.
S
= n
n
(
+
(
n
+
n
1 ) ( 2n
1)
+
A
rt
[
6
72
r
a
y
m
To fi n d th e n u b er
i d th e b a s e of wh ich i s a n
m
.
u
n
h
h
o
f
sh o
eq u
i la tera l tri an gle
t
a rra n
S pp o s e t a t e a c S ide Of t h e b a s e
b e r O f s o t in t h e lo e s t la e r is
um
h
n
t
e e
S pp o s e t a t e ac S ide Of t h e b a s e c o n t a in s n S o t ; t e n t h e
2
in t h e n e x t it is ( n
b e r O f s o t in t h e lo e s t la e r is n
o n , u p to
a s in
l
s ot
an d s o
e
a
t
t
h
e
t h e n ex t ( n
g
in
p
m
l
t
p
co
h
at
th e
+
(
y
(
1
1)
n +
u w
h
rit e n
1, n
2,
I n t is re s lt
n
b e r Of s o t I n t h e 2 r d, 3 rd,
um
n
fo r
la
i
S
(
n
7 a
pg
To fi n d the n u
id th e b a s e of wh ich
m
.
L et
re s e c t i
p
m
an
m
an
v ly
e
d n b e th e
o f the b a e
Th e t o p la
n + 1 S ot
;
d
h
so
n
s
y
er
J
:
71
2
(
1)
n +
2
(
n
n
y
s
h
o
t
in t h e
n ex
,
d we t
an
e rs
hu
w
est
s Ob
t a in
.
2)
+
A
rt
[
mb
f ho t
rec tan le
g
er
is a
o
a rra n
s
e
g
d in
a
u mb
er o
o
f
a
la
er
th e
n
a
er
th e
n
er
th e
n
la
co
m
l
t
p
e e
.
f sh o t
t h e lo n g
In
an
d
s in
l
e
g
u mb
u mb
ro
w
Of
m
(
3 (m
m— (
er
is 2
n
2)
er
is
n
3)
er
is
on
in t h e lo
th e
s
h
o rt s ide
.
y
t l y
n ex t
h
en
t
En )
7
f
c o n s is t s
in t h e
e e
.
6
73
mp l t
co
)
n
2
h
s
a
n
n
is
w
ta in
co n
in
ged
y
u mb
n
—
m
(
n
n
)
.
n
o r
PI L E S O F S H OT A N D
S
m
(
m
(
m
(
:
(
n
55
.
+ u
2
n
(
2
n
S H E LLS
1)
;
m
(
n + n
i n co
m
l
t
p
-
)
1)
+
6
3
{
—
m
(
)
n
1}
+
1)
u +
-
2n
6
74
r
a
py
m
To fi n d th e n u b er of s h o t a rra n ged i n
i d th e b a se of whi ch i s a rec ta n gle
m
.
y
u mb
um
.
y
t l y
t l y
I n t h e t o p la
in t h e
n ex
in t h e
n ex
d
so
e e
.
L e t a an d b de n o te t h e n
t o p la e r, 7 7 t h e n
b e r o f la
an
an
on
in t h e lo
er
th e
w
e st
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56
HI
GH E R
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Fi
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t
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34
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2
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t
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e
C H A PTE R
VII
.
S CA L E S O F N OTA TI ON
76
y um
h
u
Th e o rdin a r n
b e rs with wh ic we a re a c q a in t e d in
i
e t ic a re e x p res s e d b
of
l
t
l
e an
es o f
r o f 10
o
e
y
p
p
hm
.
.
A rit
fo r in s ta n c e
2 5= 2
x
m s
mu
w s
1 0 + 5;
10
m
h mth d f p ti g u mb i ll d th m
f
t ti
i
id t b
di
d
th
f h
y s l
ymb l mpl y d i h i y t m f t t i
Th
th
T is
e
o
o
den ar
ca e o
c a le
e s
n in e di it s a n d
g
s
.
re s e n
re
no
a
e
o
e rs
n
t en
o n , an
o s e
z e ro
n
s
t
n
s ca
sa
o
s s
e co
e
e ra
e
s e
o
on
n o
t
x o
on
a
o r
a re
e
e
.
um
m
h
hu
m
h
n
a
n as
e r o t e r t an ten
I n lik e
an n e r a n
b
b
e
t
a
k
e
y
y
s if 7 is t h e ra dix , a n
t h e ra dix o f a s c ale o f n o t a t io n ; t
b er
3
2
x 7
3
re re se n t s 2 x 7 + 4 x 7
5
an d in
e x re s se d b
24
5
3
;
p
p
y
t is s c a le n o d igit ig e r t a n 6 c a n o c c r
h
h h
wh
h
u
um
.
v
y
w h
o s e ra dix is de n o t e d b
A ga in in a s c ale
h
t
r
e
ab o
e
y
3
n
b e r 2 4 53 s tan ds fo r 2 r
5r 3
M o re ge n e ra ll , if in
t h e s c ale wh o s e ra dix is r w e de n o t e t h e digit s , b e gin n in g
it
’
t a t in t h e
n it s
l
ac e, b y a
a
a
t
e n th e n
b
r so
e
p
ill b e re p re se n t e d b y
or
ed
um
h
.
f m w
n
,
wh th
wh i h y
+ a
fi
e c o e f c ie n
e re
an
c
on e o r
h
f m
ts
m
H e n c e in t is
ra n
i
n
ro
O
t
o
g g
77
um
h
u
m
a
n
,
o re a
s c a le
r
1
ft
r
_
2
”
M
—l
‘
a 7
a re
er
th e
n
1
t h e digit s
a re
a
,
o ,
in t e ge rs , a ll le s s t
a
b
e z e ro
y
fi m
rs t
a 7
r
h
an
r, o
f
.
in
n
u mb
e r,
t
h
e ir
v lu
a
es
.
y
y u
y
y u
Th e n a e s B in a r , Te rn a r , Q a t e rn a r , Q in a ry , Se n a ry ,
S e p t e n a r , O c te n a ry , N o a ry , D e n a r , U n de n a ry , a n d D o de n a r
a l e s t wo ,
a re
s e d t o de n o t e t h e s c a le s c o rre s o n din
t
o th e
p
g
O f t h e ra dix
u
.
y
-
m
.
u
v u
y
58
H I
GH E R
AL
GE BR A
.
h
ym
u
de n a ry ,
s c a le s w e S a ll re
i
re S
l
b
o
s
q
h ic h a re gre at e r t a n n in e
t o re p re s e n t t h e digit s
I t is n s a l
ra dix t we l e ; wh en
t o c o n s ide r a n y s c a le h ig e r t an t at wit
we s a ll e plo t h e
n ecessar
b o ls t, e, T a s digit s t o den o t e
’
’
‘
t e n , e le e n an d t wel e
I n th e
un
w
h h h
y h m y sym
v
v
i
l
w
It i
l
t
h
f
i
y
y
h
t
p
h
h
v
.
u uu
.
s
ymb l
s
o
es
t fo r
n o
or
ec a
t
n o
o
ce
b u t fo r t h e
ten
y
ra dix
in
it sel
at
f
e
v y
er
s c a le
1 0 is t h e
.
m m
h
u
v
m
f m
w
Th e o rdin a r o p e ra t io n s o f A rit h et ic
a
b
er o r
ed
e
y
p
in d t at t h e s c ce s s i e p o e rs o f
in a n y s c ale ; b u t, b ea rin g in
t h e ra di x a re n o lo n ge r p o e rs o f t e n , in det e r in in g t h e ca rry in g
a
h
a
l
i
u
u res we
s
n
o
d
id
b
t
n
b
t
b
h
r
d
i
x
o
f
s
c
e
t
t
e
e
t
e
t
e
,
g
y
y
in q e s tio n
78
fi
.
mu
u
m
w
v
.
m
E x a p le 1
I n th e s c ale
ltip ly th e iffe ren c e b y 2 7
mu
d
.
o
h u
e ig
f
t
s
b trac t 3 7 1 532 fro
m53 02 25
,
an
d
.
53 022 5
3 7 1 53 2
1 3 647 3
27
1 3647 3
1 22 6 2 3 5
2 7 51 6 6
4 2001 1 5
um
k m
hu h v k
wh h v
h
h
m h wh h v
m wh h v
i i m
u i
w h v
w
dw
w h
mu
i
m d
dw d
u
dd i
w h
dw
E xp la n a tio n
2 we
t a e 3 fro
ic
6 fro
te n ,
.
A ga
.
ltip ly
n
,
ere fo re
t
e
n
u
A fter th e fi rs t fi g re o f th e s b trac tio n , s in c e we c an n o t
te n ,
i c lea es 7 ; t en
add 8 ; t
s we
a e t o ta e 3 fro
ic lea es 6 ; an d s o o n
lea es 4 ; t en 2 fro
e ig t
pu t
o
gby 7,
3 x 7=t
n
n
ty
5 an d c arry 2
o n e=
2
7 + 2 = fi fty
o n e=
6
o
7
n
3
I n th e
an
c arry
it
a
on
en
.
N ex t
pu t
6
x
an
d so
on
ere fo re
t
m
pu t
n
,
x
8+3;
ltip licatio n
til th e
,
3+6=
e
a e
e
1
n
o
d
an
n in e =
c arry
1
1
x
8+1
.
S i ilarly
an
d
an
d
so o n
E
xa
.
m
pl
e
2
Dvd
i i
.
E xp la n a t io n
we p u t
dw
o
A ls o 8
h
we t
n
x
1
T
erefo re
e
Sin c e 1 5
.
an
d c arry 8
e= one
pu t
l 5e t2 o b y 9 in th e
9 ) 1 5e t2 0
dw
o
n
1
x
v
T+ 5
s e en
teen
.
hu d d
n
e an
s cale o
re
d
an
c arry
d
8;
v
se
en
an
d
=e
so
on
.
wv
ft
1
el e .
x9
8,
s co
p lete
.
60
H I
v
D i ide N b y
t
r,
h
en
GH E R
th e
n
u
u
s
u
hu
T
s a ll
v
c cessi
E
xa
h
th e
re
v
i
u
q
l
m
p
e
1
u
q
re
d th e q
an
o ,
mi
a n
t ie n t
o
t ie n t is
.
de r is
a
a
o
u
2
;
.
mi
digit s a o a , ,
a re de t e r
b y t h e ra dix o f t h e p ro p o s e d s c ale
.
n e
d by
.
d
E xp re s s th e
.
th e
er
re d
di is io n s
e
d
2
r,
fu h
.
+ a r + a,
+
I f t is q o t ie n t is di ide d b y
if t h e n e x t q o tie n t
rt
n t il t e re is n o
an d s o o n ,
u
de r is
a n
r
—
n
l
v
G E BR A
mi
"" 2
a
h
re
AL
en ary n
um
b e r 52 1 3 in th e
s c ale o
v
f se
en
.
5
2
1
2
hu
T
an
s
d th e
um
n
u d
is 2 1 1 2 5
b er re q ire
m
E x a p le 2
.
m2 1 1 2 5 f m
Tran s fo r
.
1
v
s c ale s e
ro
s c ale e le
to
en
v
en
.
) 2l l 25
e
c
th e
re q
h
h
t
xa
s c ale
7 + 1= t
m
pl
3
an d
e
ten
dvd
w
dw
du
we p u t
erefo re
E
x
,
.
I n th e fi rs t lin
.
x
i i in g b y
ere fo re o n
N ex t 4
b er i s 3 to t
n
E xp la n a ti o n
t
3
u d um
ire
o
) l 244
e) 6 1
c
w k
dw
e o
or
7 + 1 = fi fte e n
we p u t
o
1
n
an
1
=
d
x e
+4;
c arry
4
.
x
ty n in e
2
e+ 7 ;
2 an d c arry 7 ; an d
en
n
f
Re
c e 7 2 1 5 fro
verI fy th e re s lt b y
u
.
7 21 5
m
so
0 11
w
v
w k
s c ale
or
t
.
el e
i n g in th e
s cale
ten b y
s c ale t el e
to
21
7
0 5
wv
w k
or
in g in
.
12
1 24 01
hu
T
th e
s
re s
u
lt is 1 2 4 01 in
h
wv
e ac
c as e
.
m
x
E xp lan a tio n
7 2 1 5 in s c ale t el e
e an s 7 x 1 2 3 + 2
x 12
5 in
Th e calc latio n is
s cale t en
o s t re a dily e f
fe c te b y
ritin g t is e x re s s io n
p
x
I n th e fo r
x
x
t
s we
ltip ly 7 b y 1 2 , an d
a dd 2 t o th e p ro d c t ; t en we
ltip ly 8 6 b y 1 2 an d add 1 to th e
ct ;
p ro
t e n 1 03 3 b y 1 2 an d add 5 t o th e p ro
ct
u
.
.
h
m
u
h
m
mu
du
.
d w
hu mu
h
du
SC A L E S
f
h
o r N O TA TIO
hv
y
u
a e o n l
H it e rto w e
80
dis c
a
a ls o b e e x re s s e d in a n
ra c t io n s
p
y
y
m
.
2 5 in
s c a le
2 5 in
2
d
s c a le
sse
yd
o rdin a r
s c a le r
wh
or
s c a le r
de n o t e s
s
a
ra c t io n s
t h e radix p o in t
-
lo
5
6
2
6
m my b
o re
.
e z e ro
a
To
n
en
l les s t
ra c t io n
b
b
e
th e
h
an
,
f
ra c tio n
an
7
v wt fi
M u lt ip ly b t h id
ha
e no
o
o
s
n
d
in
th e
r
H
th e
f
i
u
q
re d
r, o
f
s
to
4
7
u
o
f
in t
ra c t o n s
in
wh i h
c
e
an
y
o n e
an
y p
ra dix o
ro
p
o s ed s c a le
Q
a
e e
b
f t h e p ro po s e d
f mh
t
ro
e
+
es o
+
”
7
f b , , b , , b a,
a t io n
b
+J
-
,
by
r
; t
h
en
L h
2
hv
h
u
.
‘
v lu
h
u
q
u
M ltiply
at
o
t
an
digit s b e gin n in g
b l is e q a l t o t h e in t egral p a rt
ra c t io n al
a rt b
w
a e
e
,
p
y R
en ce
t
b
r
ft
F=
r
‘
d th e
es o
re
b
6
We
s
u
h
s d h p
uhf i
.
f
hu
r
al
ra dix
n
i
v
e
g
v f
L et b , , b
le t ; t h e n
t
but
;
5
f m
y
in t e ge rs ,
e rs
5
-
.
u mb
.
exp res s a
L e t F b e t h e gi
c a le
le
f n o t a t io n
10
2
o
in a
o r
a n a lo o
g
a re c a lle d radix frac t io n ,
Th e ge n e ra l t p e o f s c
is
a re
o
5
2
de n o t e s
e x p re s s e d
t
b ,, b , ,
e re
81
s
ec
c a lle d
is
hu
iml f
ra c t io n s
wh
2
r
F
61
.
t e n de n o t e s —
s c a le s ix
0 In
N
f
o
f rF
an
d, if w e de n o t e
i
s t h e in t e gra l p a rt
i
re , 6
b
n
a
b
e
o
r
t
e
s
g
;
,
y
2
o f
o f
an d si
ila rl b y s c c e s s i e
ltip lic a t io n s b y r, e a c
t h e digit s
a
b
ra c t io n e x p re s s e d in t h e p ro
e
o
n
d
a
n
t
h
e
d
,
y
o s e d s c a le
p
a
m
.
a n
m y
fu
v mu
f
h
62
HI
GH E R
AL
GE B R A
v mu
u
.
lt iplic a tio n s b y r an y o n e o f t h e
I f in t h e s c c e s s i e
i
n a t e s a t t is s t a e
ro d c t s is a n in t e ge r t h e p ro c e s s t e r
a
n d
,
p
g
n it e n
t h e gi e n
rac t io n c an b e e x p re s s e d b
a
r
b
e
o f di it s
y
g
B u t if n o n e o f t h e p ro d c t s is an in t ege r t h e p ro c e s s will n e e r
t e r in at e , an d in t i c a s e t h e digit re c r,
o r in
a ra dix
g
s t o a re c rrin
d
al
ra c t io n an alo o
e ci
g
g
u
v f
m
f
m
E x a p le 1
1
m
2
d
13
13
as a ra
1
x
2
6_ 3
We
s
frac ti o n
4
5
1
6
6
63
2
m
th e in tegral
3
6
4
.
fro
Tran s fo r
re at
3
.
4 51 3
.
.
2
1
m
mu t t
s cale s ix
3
x
4
E x a p le 2
.
4
1
i re
u f m
x3
7
8
u d
x
v
.
8
7
re q
um
fi
ix frac tio n in
16
th e
h
s
u
1
1
E x pre s s
.
u
hs
u
m
m
h
s c ale e ig
d th e frac tio n al p arts
an
s c ale
t to
fi ve
.
s ep arate ly ,
24
-
5
5
5
4 04
5
-
h
A fter t i s th e
n
um
b er i s
d
igit s in th e frac tio n al p art
2 1 2 3 4 0 12 4 0
rec
u h
r;
en ce
th e
re q
u d
ire
.
m
82
I n a n y s ca le of n o ta tio n of whi ch th e ra dix is r th e s u
,
of th e digi ts of a n y wh o le n u b er di vided by r — 1 will lea ve th e
sa
e re
a in der a s th e wh o le n u
b er divided b y r
1
7
L e t V de n o t e t h e n
b e r, a o , a , , a 2 ,
a
t h e digit s b egin
’
n n
w
i
t h t at in t h e
n it s
l
g
o f t h e di it s
p a c e, a n d S t h e s u
g ;
.
m m
I
t
h
i
en
u
N
N —S
z
a,
(
r
a
o
a r
,
um
a r
m
m
.
n
m
2
"
a r
2
— l
n
)
+
+ a
(
"“
r
;
—1
)
+ a
”
r
—l
)
.
S CA L E S O F N OTA TIO
Now
v yt m
er
e
the
o n
er
V
1
r
H
su
so
mi t
u mb
e
n
S
o
wh i h p v th p p i i
l
w ill b div i ib l b y
er
ro
c
g
er
digits
s
1
in s c a e r
is di is ib le b y
en c e a n
m f it
e
v
1
r
t e ge r ;
In
1
I is
83
v
N
t h a t is ,
e re
an
63
.
d s ide is di is ib le b
y
an
S
x
wh
h h
ig t
r
N
es
e
s
e
l
r
t
os
ro
e
on
.
1
r
wh
en
th e
.
f m
v
w
v
m m
u
w
u y mu
B y t ak in g r = 1 0 we le a rn ro
t h e ab o e p ro p o s it io n
t at a n
b e r di ide d b y 9
ill le a e t h e s a e re a in de r a s t h e
“
su
o f its di it s di ide d b
9
T
r
l
k
n
h
e
e
o
n as
c a s t in
out
g
y
g
”
t h e n in es
fo r t e s t in g t h e a c c ra c o f
lt iplic a t io n is o n de d
o n t is
ro
e
r
t
p p
h
um
.
m
h
v
y
v
.
.
u l my b hu
tw
u mb b
du t b y P; t h
Th e
e
r
Le t
t e ir p ro
h
o
a
e
t
e rs
n
re
l
a in e d
p
p
re s e n
te d b y 9 a
b
an
d 9c
+
d,
an
d
en
P
e n ce
s ex
e
c
P
H
fu
h as th e
:
8 la e
9bc
sa
m mi
re
e
a n
9ad
de r
bd
.
bd
as
an
d t
h f
e re o re
th e
9
su
h en di ide d b y 9 , gi e s t h e s a e
of th e digits
o f P,
re
a in de r a s t h e s u
of th e digits o f b d,
e n di ide d b
I
9
f
y
o n t ria l t is s o
ltip lic a tio n
ld n o t b e t h e c a s e , t h e
st
a e
I n p ra c tic e b a n d d a re re a dil
b e e n in c o rre c t l p e r o r e d
o
n d
b e rs t o b e
ro
th e
o f t h e digit
o f t h e t wo n
ltiplie d t o ge th e r
9
m
m
h hu
y
f m
f
u
mu
m
f m
su ms
v
.
wh
mu
v
s
m
v
mu h v
y
um
.
.
m
um
E xa p le
Th e s
2 1 , an d 3 4
8, 3,
igits ; t
are
d
w
.
C an th e p ro
d
v
wh
hv d
du
ct o
f 3 1 2 56
mu
an
d 8 4 2 7 b e 2 6 33 9 53 1 2 ?
d mu
d
wh h
ltiplican ,
f th e igits o f th e
re s pe c ti ely ; again , th e s
s o f th e
ic
e n ce b d = 8 x
an d 7 ,
iffe ren t re ain ers , 6
s we
a e two
s o
hu
is in co rrec t
um
m
d
du
ltip lie r, a n d p ro
c t are 1 7 ,
b e rs
igits o f t es e t ree n
O f th e
h as 6 fo r th e s u
ltip lic atio n
an d 7 , an d th e
h
h
mu
m
u
m
.
m
84
If N den o te a n y n u b er i n th e s ca le of r, a n d D den o te
i
i
t
s i n th e
o
d
h
u
t
e
u
i
v
t
e
n
t
h
e
s
s
w
th e dfi
r
en
c
e
o
se
d
s
t
i
e
b
e
e
z
s
o
,
g
f
,
fi
pp
p
—
u lti le o
D
s
a
l
a
N
o
N
i
d
d
a
n
t
v
e
n
ces
h
en
D
r
d
h
e
e
t
+
o
f
;
p
p
l
r
.
.
,
m
m
64
HI
Let
th e
a
u
in
n
a
, ,
a
2 ,
’
it s p la c e ; t
h
GH E R
AL
GE B R A
.
n
a
+ a,
o
dl
a
1
+
)
(
2
ill b e
o n
t h e rig t
T
s e
o dd o r e e n
m
d t h e la s t t e r
i
s
as n
a c c o rdin
g
1
di is ib le b y r
an
v
—
N
(
a
h
—
a
O
,
+ a
2
l
th e
by
wh
1
r
E
d
o s e ra
L et
r
is
1
h
h
w
v h
h
d h
1
Pro e t
is gre ater t
e
.
ix
b e th e
v
s
th e gi
E
xa
ra
ix
is
at
an
t
4
m
pl
en
e
2
n
.
b e th e
a sq
en c e
So
th e
um
wh
b er is th e
In
at
s c ale
t
h
sq
’
u
‘
in t e bg e r
e
v
u
v
l
a c e s is e
al t o
p
q
0, a n d N is di is ib le
en
ra
d
ix is 4
e
m
es
E x a p le 3
b y 1 01 2 1 5
.
I
are o
f2 1
d
u d
um
b e r in
an
y
s c ale o
mu
o
um
tatio n
b e r 2 4 3 7 5 rep re sen te
d
by
i
2 '4 3 7 5z 2
r
1 2)
°
(
'
r
at s c ale
mu
mu
u se
w
th e
ill th e
48
0
4) = 0
.
f ll w i g m h
um
o
o
et
n
d
.
b e r 2 5607 b e
n o n ary n
h
hh
o
ire s cale
s t b e les s t an 9 , s in c e th e n e w n
th e gre at e r ; als o it
s t b e gre ater t an 5
; t erefo re th e
s t b e 6 , 7 , o r 8 ; an d b
y tria l we fi n d t at it is 7
re q
fn
.
e n ary n
.
wh
are n
en
it is b e s t t o
In
u
;
is th e
s c ale
(7 7
mtim
Th e
.
1
1 67
is,
H
h
en
en ce
at
th e
.
r
t
an
m
1
r
on
.
‘
L et
n
1
)
(
ri
t
i
s
g
-
é
hu
"
r
in t e g e r
an
— +
t
er
r
o r a
hu v y t m
er
(
3
a
.
m
pl
xa
)
+ a
.
f (b
t h e p ro p o s it io n
m
su
— 1
9
r
I f th e s u
o f t h e digit s in t h e
o f t h e digit s in t h e o dd p la c e s , D
.
+ a r
7
a
0
4
r
C OR
.
—
-
N
es
h w
(
1
O
ro
at
e n ce
( L — (b
c
+
r
v
r
wh i h p v
t
en
+ a
JV—
wi h th
de n o t e t h e digit s b e gin n in g
a
h
.
d
e x p res s e
um
u d
b er app ea rs
re q ire
s c ale
N
du d
SC A L E S O F
m
w k m
d wh v u
O TA TI O N
By
o r in g in th e
E x a p le 4
ol
e is 3 6 4
o se
rec tan g lar s o li
o s e b as e is 4 6 s q ft 8 s q i n
u
v um
wh
Th e
264 7 34
.
l
cu b
e
o
is 3 6 4 44443 c u b
a e
fi n d th e e ig t o f a
ft 1 04 8 c u b in , an d th e are a o f
.
.
.
.
ft
.
.
wh h
ic
,
exp re sse
d
in th e
wv
s c ale o
f t
el e
is
.
is 4 6g
wh h
dvd
ic
1 s q ft ,
t ere fo re to i i
.
.
hv h
We
cu b
h h
s c ale ,
e n ary
.
.
ft
.
Th e are a
.
.
o
65
.
e x p re s s e
d
in th e
s cale o
2 6 4 7 3 4 b y 3 t 08 in th e
e
°
wv
wv
f t el
s c ale o
e
is 3 t o 8
'
el e
ft
.
.
22 t 4 8
hu
T
h h
e ig
th e
s
3 627 4
3627 4
t is 7 ft 1 1 111
.
.
E XA M PLE S
VI I b
.
.
E x pres s 4 9 54 in th e
1
.
2
.
3
.
4
.
E x pre s s 6 2 4 in th e
E xpre s s 1 4 58 in
.
7
.
8
.
Tran s fo r
9
.
Tran s fo r
10
.
Tran s fo r
11
.
Tran s fo r
12
E xp res s
13
.
Tran s
14
.
.
f
m
.
re e
.
.
fu
s c ale
e n ary n
o
II
)
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.
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17
.
Tran s fo r
sca e
‘
o
s c ale
ro
o
e
ro
ro
Fi
m
t er
.
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21
.
In
.
H
.
e
A
.
rac ti o n
s c ale
ten t o
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1 552
26 26
s c ale
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s c ale
.
.
.
.
n o n ar
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y
to th e
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.
o c te n ary s cale
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vu lg f
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.
a
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20
s
o
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s e p t e n ary s cale
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n ar
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in t h e
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.
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n
e
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as a ra
E xpres s th e
.
e rs Of 1 0
o sca e e
m2002 1 1 f mth t
m f mth du d
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w
.
e n o n ar
er
n
ro
m1 7 1 56 2 5
w
fi ve
sca e e e
o
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en ar
eee
o
sca e n n e
ro
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e s e n ar
ro
Tran s fo r
.
s cale
to
r
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er
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ro
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s cale
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l
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m6 t1 2 f
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m2 1 3 01 4 f mth
t
h
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m2 3 8 6 1 f m l i t l ight
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m4 00803 f mth
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y
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206 6 51 52 i p w
p
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mttt f m l tw lv t th mm
15
19
f fi ve
umb
m l tw l
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E x pres s th e
f s e ve n
s c ale o
w
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lo
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E x p re ss 538 1 in po e rs
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18
s c ale o
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th e de n ary
th e
d
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'
n
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1 82 de n o te
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v
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2 22 ?
0 302 ?
5
66
GH ER
HI
AL
GE B R A
.
h
d h d i f h l i whi h
u
p
q
f
2
d by 7
l i
7 d
I wh
23
l i wh i h h
Fi d h di f h
u mb d d b y
24
i
i hm i l p g
i
7 9 6 9 8 07
i g m i
th
l
di f i
25 I wh t
i
?
g
p
h
l
f i
i wh
u mb
i i
l w ill i
Th
26
d by 7 6?
b d
l
i
i
i
h
f
t
u
v
wh
h
i
8
Sh w h
27
q
y
p
di g
h igh
i
l
i
i
f
u
wh
d
Sh w h
28
q
p
y
i
lw y
h
h
d by h
d h
i g
p
qu
m f u d igi
l
b
i
i
i
f
u
wh
d
i
P v th
29
y
p
h
h
g
lb
mu b u d
30 F i d wh i h f h w igh
w igh
Fi d wh i h f h w igh
lb
m b u d
7
31
f
w igh th u d lb t m h
h ki d b i g u d
i
l h
i h
i
b
y
i
ub i v y l i whi h th
S h w th
7
32
p f
di g
h v
i
h
d i y l u mb will b d ivi ib l b y
P v th
33
u mb f md b y i l h d igi i divi ib l b y igh
if h
h
l
h
u
i
f
i
wh
P v th
f
O
O
O
34
q
q
i
i
t
h
v
g
y
q
b
N b
k i h l
b
w um
d
N
35 If y u m
d
f i
d igi i y w y h w h h
b f md b y l i g h
d iff
N
d N i d iv i ib l b y
b w
umb h
v u mb f d igi h w th i i
If
36
d ivi ib l b y if h d igi q u id i f m h d h m
h
d i y l S b h m f h digi f umb
If i
37
d S b
h
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HIGHER
74
I MA G I
N
G
A L E BR A
Q
A RY
UA
N
.
TI TI E S
.
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m
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y
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t
J
J
y
p
q
d th i
f qu t u
mth mti l i v tig ti
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lu b l
l ds t
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i wh t
w
i
t
v
di l ig i
W h th qu tity u d t h
g
th
ymb l J i di ti g p ib l ith mti l
id
l g
m
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y
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a
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d c o n e n ie n t t o in dic a te t h e
b y t h e p re s e n c e o f t h e
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be
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s h all a l
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95
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lly
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94
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e
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th e p ro d c t Ja
an t it
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q
T
a n
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is ,
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b
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o rt an c e w ic h
p
a
m
m
ic h
f t h e s ign s
e s i n is t o b e t ak e n
g
t h e re is o n e p o in t o f i
ra r
os
c o n s ide r
h
HIGHER
76
AL
GE B R A
.
m
m
a y b e equ a l i t
i agin a ry exp res s io n s
is n ec ess ary a n d s ufi cien t th a t th e rea l p a r ts s ho u ld b e equ a l, a n d
th e i a gin a ry p a rts s h o u ld b e equ a l
Th u s i n
th a t two
o rder
m
.
N
.
y
N
h
m
y
hy
W en two i agin a r
D E F I I TI O
a rt
t
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p
1 00
on l
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j
m
.
y
c o n u gat e
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b
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b
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b
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it
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ju g t
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ro o
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+
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a
b J;
a
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S
UR DS
IMA
AND
GINA R Y
U A N T ITI E S
Q
77
.
Fo r in s ta n c e
0
4
d
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A rt 9 7 , we s e e t a t th e s u , difi eren ce,
u o tie n t o
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HI G HER
78
Sin c e th e p ro
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h
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4
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/
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1 ;
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an
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d
an
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s
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1
Si ilarly
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2 _
x
a x
3
mu t t
or x
e Of
J J
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GE BR A
7
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wh
we
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e ro o ts are
is
AL
1
2 a (1 i
J
J
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.
H I G HER
80
A ls o w
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is , th e
at
A gain
t
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o
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at o n
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3
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y
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f
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1
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W e n o w s e e t a t e e r q an t it h as t re e c b e ro o t s ,
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s
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at
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I n th e p ro
GIN A R Y
I MA
AN D
a
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E XA M PLE S
VI II b
.
1
u
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) (1
—
m) ( l
4
—
u
f
n
ib
+
ity , p ro
-
.
i
—
1
w
a
ib )
2
v
e
— (1 +
2
—
( l co l w )
25
1—w
(a
e ro o t s o
.
5 =
w
)
9
co
o
2
( + 5w l
-
1
(
+w
co
Pro
) (1
2
v h
e
t
+2
s
th e t
are
2
’
(O
2
00
4
) (1
to 2 h
fac to rs =
2
2n
at
3
3 e ez
If
(N
=
x = a
e w)
+ b, y
( 93 + re
= a co
+b
m
2
a co
,
2
+
b w,
at
1
( )
3
=
x z
a
y
+b
3
.
(2)
(3)
31
s
.
hw h
e
t
3
x
3
+y +
3=
z
3
If
at
cx
(
a
2
2
+b +
2
c
— bc —
+ by + az
—
ab
ca
) (
x
2
=
Y,
2
+y +
=
2
2—
X +I
2
’
2
— ex — x
z
y
+Z
z
y)
YZ — XZ
HI
AL
I n A rt 1 1 1 le t t h e t wo
1 13
s o th at
,
b
J
GEBR A
ro o
.
.
B
GH E R
2
e de n o te d b
b
y
( )
t s in
—
4ao
b
+
.
b—
2
v
J
o
o
— 4 ao th e
I
f
b
1
(
( )
a re re a l a n d u n e
an d
B
I f b — 4 a o is
2
2
( )
u
re d
h
i
n
i
s
t
g
c in
3
( )
Ifb
If b
4
( )
un e
u
q
2
2
a
z e ro
n e
4 a o is
u
o r an
,
re a l
a re
ga
at
g
p
a
d Ba re i
a
u
d
e, ( 1 an
er ect sq
S
e s Of x
at
t
e
a re , a a n
v
e,
e
y
n ar
B
an
u
q
d
e ach
a l,
u
n e
i
a re ra t o n a
u
q
l
a
l
.
an
d
a
erefo re
th e ro o ts
xa
m
pl
Th e
co n
e
d
2
.
u
u
al ro o
x
S
.
hw h
e
t
x
v
ts gi
(k
es
2
576
an
y
at
2
-
th e
ro o
ts
2p x + p
2
.
=
0
c an n o
h
t
hv
.
e
Sin c e
by
a
t be
s atis fi e
d
at
.
0 h as e q
9k
x
4
h
1
‘
01
O f th e e q
u
atio n
2
2
r
+
q
q
2p )
4 (p
2
4
to
(q
2
ce s
?
ddit io n
+
J
b
ts , fi n d
.
a
b
al ro o
0,
v d dud
2
u
9 k,
.
1 14
.
— 20
2)
ill b e ratio n al p ro i e
Th e ro o ts
u t t i s e x p re ss io n re
r
B
e
a
r
f
c
t
s
e
e
q
p
H en c e th e ro o ts are ratio n al
.
u
) (k
are ratio n al
w
u
v
.
2 (k
2
k= 4,
3
s
n ary .
a
e q atio n
itio n fo r e q
(
0
e
o
‘
4
-
0 —4
m
pl
re
so
k
xa
u
= 2 b
,
mgi
i
are
I f th e
o
u
.
b — 4 ac =
a
is p o s it i
f t h e ro o t
Of
t s o l i ng t h e e q a t io n
n at
3 —
i
o
2
x
6x + 7
e q at n
th e
2
we
d
an
mgi
v
f
i
h
m w h
m
hw h
u
m
u
H ere
E
B
d
y
.
E
)
ra dic a l
.
E x a p le 1
b y an y real val
h
de r t h e
.
B y a pp l in g t e s e t es t s t h e
i
t
a
b
e de t e r in e d
a drat ic
y
q
T
n
6
al
u
.
ts
t
an
to
c as e
4 a o is
l
u
g
i
t
u
y
q
l
u
q
re s
n
2
a
g
2a
f ll w i
th e
e
d
an
b — 4 ac
2a
t h e n we h a
a
—
4ao
—b —
2a
J
b — 4 ac
z
2
is
q + 2 qr
or
4 (q
a
an
mu ltipli
OF
Q U A D R A TI O
4a
b)
3
th e
s
e re s
In
u n
a
ity ,
e
u l my
d
t
i
u
q
ts
a
ra
u
q
a ls o
a
c e
i
be
u
q
u
i
i
h
t
e
ro d
p
( )
u
u
u
1 15
e
u
q
(6
e x p re
atio n
c alle
d
f m
or
si
s e
f ll w
ed as
wh ere th e
t
ro o
ss
u
q
is
fi
B
al
hd
a t io n
x +
0
x
f
t
o
m
th e fi rs t ter
fi
c o ef c ie n
t
o
f
x
wit
h
m
t o t h e t h ird t e r
m
o es n o
t
co n
.
tain th e
uk w
n
no
n
.
d
r
e
a
.
cien
i
tt
w
m
b
y
g2
m
u
i
m
l
t
b
d
y q
p
y
m
f
u
f
t
d
t
p
)
(
f m( 1 ) w h
2
s
t o th e
u
q
an
,
o
co e
al
e
mwh
o
th e ter
ic
th e a b s o lu te ter
Sin c e
.
th e
Cb
c t Of t h e ro o t s
N OTE
I n an y eq
q an tity is fre q en tly
.
c
i
m
h
b
at o n
2
2
in t h e
at o n
i
h
t
e
s
u
Of t h e
()
s i n c an e d
g
g
its
85
.
4 ao )
2
(b
4a
writin g
U A TI O N S
EQ
we h a ve
c a t io n
d by
Y
TH E O R
TH E
en
e
H
e n c e an
x
A ga in ,
E
xa
Th e
2
e
u
eq
1
ra
.
atio n
w
—
x
a
)(
th e
or
eq
is
mt h
e
o
h
en
d
.
th e
ro o
atio n
(
ts
a re
e
an
x
B)
=
O
ro o
in t h e
ts
f m
or
0
0
a t io n
o se ro o
3)
x
W
x
—
y f m qu
u wh
m
e a s il
Fo r
c
ro
x
re ss e d
a ve
e
n o
s
ro o
o
e ex
a so
a
c
(
a
m
pl
su
ro
my
We
a
2
(
x
x
ts
w i h giv
t
are
3
an
ro o
en
d
2
ts
.
.
2 ) = 0,
6
0
irra t io n a l it is
.
e a s ie r
to
u se
the
f ll w i
o
o
n
g
86
HI
m
E x a p le 2
hv
u
We
th e
by
u
s in
t
s
E
u
p ro
mul
g fo r
mf
du
O f th e
mh
p re sen t
f m
u
mu
wh
an
3
J
d2
.
0,
.
h t u di
w ith h
u
a
se
n
t
mpl 1
m giv
Exa
re e o r
e
o re
o
f
en
.
xa
m
pl
e
1
mth
Fo r
.
u d u
eq
atio n
e eq
ere fo re
th e
eq
u
x
ts
by
—
are
eac
2,
h
3,
o
an
d
g
w
f th e fo llo
in g
su p
.
7
g
e
s
(
d
+ 3 : 0,
x
mu t b
m
<
atio n
o s e ro o
satis fi e
be
st
atio n
x
h
1
artic le
u
2 + ,J3
are
1;
s to t
e t o d a n a lo go
By a
o r
an e
a t io n
a rtic le w e c a n
q
Th e req ire
p o s itio n s :
t
4x
2
ts
.
ts : 4 ,
c t Of ro o ts
x
2
( )
G E BR A
o s e ro o
ro o
O
is
a
AL
wh
atio n
su
atio n
eq
.
ro o
e eq
a e
116
las t
th e
mth
Fo r
.
GH E R
g
( )
a
o
-
(5x
x
2
—
2x
5x
3
m
E x a p le 2
Th e
h
t
eq
e re fo re
u
mth
Fo r
.
atio n
e eq
h as to b e
u
wh
atio n
d
by
a,
x
s atis fi e
x
=
+
a
o s e ro o
= — a,
ts
are
0,
i
a
x
it is
(
x
x
x
bx
u
4
(x
6x
)(
x
— a2
3
—a
)
) (bx
a
2
bx
2
0
a ex
.
m m
m
m
y
o st i
Th e re s lt s o f A rt 1 1 4 a re
o rt a n t , an d t h e
p
ro b le
a re
so l e
s c o n n e c t e d wit
e n e ra ll
s f
fi
c ie n t t o
t
h
e
p
g
ro o t s o f
a dra t ic s
n s c
e s t io n s th e ro o ts sh o u ld n ever b e
I
q
q
l
c o n s idered s i n
l
u t u se s o
d
b
e
a
d
h
e
o
f
b
t
r
e
l
a
t
i
n
e
o
s
Ob
,
gy
rit in
t a in e d b y
d
o
n th e s u
o f t h e ro o t s , an d t e ir
r
o
t
d
c
g
,
p
in t e r s o f t h e c o e f c ie n ts o f t h e e q at io n
117
.
u
w
m
E
( 1)
0
xa
8
m
pl
e
1
2
—
2
4 19
( )
We
.
hv
a
y u
e
.
.
fi
If
.
d
3
“
a
.
3
an
w
d
v
uh u
hu
m
B are
th e
u
ro o
ts
B
p,
+ l3
a
:
Of x
h
h
u
.
2
fi n d th e
v u
al
e Of
88
HI
GH ER
G
A L E BR A
.
To fi n d th e c o n di tio n th a t the
1 18
2
1
equ a l i n
ax + b x + c = 0 s h o u ld b e
( )
in s ign , ( 2) recip ro ca ls
f
ro o ts
mg
.
a
n
e an
d
th e equ a tio n
o
i tu de
d
an
op
p
o s ite
.
Th e
t e ir s u
h
ro o
mi
t
s w ill b
e eq
u
al
h en c e t h e
s z e ro
mg it u d
in
a
re
é
—
A ga in , t h e ro o t s will b e
st h a e
u n it
h e n c e we
mu
y
a
h
d
or
b
p
1,
-
re
re c i
v
0
fi
i
u
q
0,
(b
n
u
dit io n
co n
a
z
f
u
pp
is
o s it e
in
if
Si n
g
0
=
.
ro c a ls
or c
O
wh
t
en
h
e ir
p
ro
u
d
is
ct
.
u
u
Th e rs t O f t e s e re s lts is o f re q e n t o c c rre n c e in A n aly ti
o re
e t r , a n d t h e s e c o n d is a p a rt ic u la r c a s e o f a
c a l G eo
e n e ral c o n dit io n a pplic a b le t o e q a t io n s Of a n y degree
g
m y
E
h
xa
m
pl
v
i
i
o
s
e,
t
p
bot
d
Fin
e.
d
th e c o n
2
o pp o s ite in
)
(
W e h ave
I f th e
li e s ign s
(1 )
hv k
a e
Al s o , s in ce
ts
ro o
are
h
v
a
+ B is p o s iti
s ign s .
,
u d
hm
b
a
v
e,
is
a
,
p o s iti
b ot
.
h
v
B is p o s iti
v h
h
at
th e
t
h
d t
e , an
e refo re
b
s ign s O f a an
an
a
a e
A ls o
Sin ce 0
hv u k
fo re
an
are O f o pp o s ite Sign s , a
d0
+ 73 h as th e
Sign
v h
u d d
i
v
B is
n egati e , an
i s p o s iti
e;
t
e re fo re
b
an
o
f th e greater
d
a
hv k
h
li
a e
H en ce th e req ire c o n itio n is t
d o pp o s ite to th e Sign o f c
IX
.
mth
Fo r
4
.
7i 2
e eq
u
at io n s
wh
5
.
o s e ro o
ts
are
+ 2 N/3 — 5
.
.
a
.
da
hv u k
a e
n
li
h ud k
h
v
h
s
o
l
b e li
dt
e
e,
ere fo re c an
n egati e , an
d
ere
.
s ign s O f a an
.
E XA M PLE S
t i t is
e s ign s
th e
at
ro o
a
dt
.
.
.
erefo re c an
d
.
I f th e ro o ts
n li e s ign s
1
( )
e
(l
n e gati e ;
d
v
a
c
B
e, d
H en c e th e req ire c o n itio n i s t
an d o pp o s ite to th e Sign Of b
2
( )
my b
2
h
at t e ro o ts O f a x + b x + c = 0
n egati e
b u t th e greater o f t e
+B
a
.
itio n t
Sign
m
db
s
h ud
o
l
k
b e li
e,
TH E
7
.
1
5 Q
t
e
1
( )
‘
l
x
.
2
( )
(
If th e
14
al es O f
.
th e
ts
o
— 2a x a 2 —
+
b
.
f th e fo llo
3
‘
— c2 =
u
at io n
x
2
at
es Of
a
.
u
al ro o
Qi
.
(a
N /3 )
b)
4
.
.
at io n s a re re al
b
h as
eq
ul
a
ro o
ts , fi n d
2
.
ts
at
2x
a
e Of
mw ill th
t
e
at
(
2
( )
2 2
a bc x
a
a
;
21
.
.
3
x
2
ts
x
24
.
If
a
o s e ro o
ts
ax
2
an
22
d B
26
I fx “
e
2
6a
u
S ign
n
e
atio n s are rat io n a l
are
a
b
2
2b
°
0
.
bx + 0
:
0, fi n d th e
v lu
a
es
h
4a w
th e
an
ro o
th e
.
i
3
:
ro o
ts
o
l
3
.
mth
f
fo r
ts
o
ts
f (x
o
a
)(
(
a rz
x
fa v + bx
2
2
,
b)
J
e
eq
d
ro o
3
.
1
en
<
ax
z
2i
1
-
(
in
wi g q u
a t io n a x
1)
2
( )
fo llo
e
a tio n
o
i
t
s
e
pp
o
cx
eq
en x
3
(a
t at t h e
x
b
2
en x
a re
v h
25
Pro
t
u
.
wh
wh
2
2a x
are
u
o
f th e
o
8x + 1 5
3
0
e Of
x
3
x
.
.
ts
19
a
e
23
.
e ro o
2
18
m)
m— l
m+ 1
bx
—c
e
2
3 a cx
ro o
d th v lu
3
t
1
t
2
+o
If a , B are th e
Fin
t
l
( )
1
a n
n
a
e eq
'
u l i mg i ud b
v h h
f th
eq
Pro
a tio n
ts ?
wh v lu
Fo r
e ro o
17
h
e
x
e e
ax
w
n
mwill th qu
( 1 3m
) + 7 (3
x —
22
wi g qu
i
,
0,
m( 2
15
2
a
12
—
wh v lu
Fo r
e eq
16
hv
2
a
x
a
9
.
89
.
.
15
hv
ro o
U A TI O N S
EQ
ih
O,
,
2
at
eq
m
.
a
,
v h
Pro
.
Q U A D R A TI c
a
2
—3
,
10
v u
OF
3i
.
13
Y
TH E O R
b)
3
.
b)
=
c=
h
2
wy
h v lu
a re al
0, fi n d t
a
e
s re al
a
.
e o
f
u
a t io n
90
27
ti
Fi
n
m
n
.
d
th e
es
d iti
th e c o n
o t er
h
F mth
or
.
e eq
f th e diffe re n c e
o
u
‘
2
D is c
u
ss
GE BR A
E
xa
e
wh
mth
ro o
ts
s ign s o
o
o s e ro o
ts
+
(m
f th e
n
)
ro o
If x is
m+
2
x
ts
f th e
o
m
um
v
hv
e ric al
a e all n
c an
L et th e gi
qu an tity , p ro
al e s ex ce p t s
be
x
2
h
multiply i
n
en
g
u p an
x
h
eq
a
e
e eq
u
u
u
are s Of
wh
a tio n
th e
su
o se
m
an
d
0
.
a tio n
u fu l
se
a
a
l
i
a t io n
c
pp
t
e
as
w
h
lie b e t
by y
te
so
.
e x p re s s io n
th e
at
een
t
2
an
d6
.
at
11
2x
3)
(x
v h
c
hv
d tran s p o s in g, we
2
2=
s ra t es
uh
d
rep re s e n
2
t
h ll b
.
v u
en e x p re s s io n
n
sq
ut
.
a re al
th e
are
f
f w
v
.
s
+B
at io n
f th e
th e
u
l
m
p
f
o
b x + c = 0, fo r
?
Th e o llo in g e x a p le ill
119
o f t h e re s lt s p ro e d i n A rt 1 13
.
t
ro o
one
.
“
2x + 2
.
at
O f ax
ts
ro o
2
30
h
AL
.
I f a , B are th e
28
2
2
ro o ts are a + B an d a
29
t
on
.
o
GH E R
HI
a e
—
2
1
x
+
y) + 6 y
(
d h mhv
v u
v d d
m
v
mu
h v
h
v
h
h
m hv
h v u
ti d t h t t h q
d
i
i
p
d
l
i
b
w
t
t
h
t
t
y
g
ud
u
mu
mu
v h
h du mu
h
w
T is is a q a ratic e q atio n , an d in o r e r t at x
al e s
ay
a e real
2—
— 11
4
o s iti e ; o r
l
f
6
s
b
e
i
i
n
s
i
i
i
n
n
t
v
i
b
4
d
a
p
)
y
g,
g
p
y
y
y
)
(
2
s t b e p o s iti e ; t at is , ( y
8g 1 2
6 ) (y 2 )
s t b e p o s iti e
H en ce
y
st b e b o t
ct
In th e
t h e fac to rs O f t is p ro
p o s iti e, o r b o t n egati e
fo r e r c as e y is gre ate r t an 6 ; in th e latte r y is les s t an 2
T erefo re
2
n d 6, b u t
n y o t er
n
n
l
i
e
b
e
t
e
n
a
a
e
a
l
a
c
a
o
t
e
a
e
y
y
-
m
h
m
.
.
.
.
a
I n t is e x a p le it will b e n o c e
2
o es n o
i
i
t
i
e s o lo n
as
8
1
2
s
o
s
y
p
y
2
u adra tic e u a tio n
o f t h e c o rre s o n din
q
y
p
gq
in
h
v
u
T is is a p a rt ic la r c a s e
t h e n e x t a rt ic le
Of
t h e ge n
e ra l
e
e
8y
p
ro
ra t c ex
u a
e
e en
12
ress o n
e ro o
s
0
.
o s it io n
p
in
v
e s tiga t e d
.
Fo r
1 20
ll
rea l va lu es
f
the
exp ress io n
2
bx
2
th e s a e s ign a s a , ex c ep t wh en th e ro o ts of th e e u a tio n ax + b x
q
a re r ea l a n d u n e u a l, a n d x h a s a va lu e l in
q
y g b etween th e
m
.
a
x
ax
m
CA SE 1
.
u
S pp o s e t
h
at
th e
ax
a re re a l
o
de n o t e t
h mb y
e
a
2
an
ro o
ts
bx
d
B
o
c
,
an
f th e
e
u
q
a tio n
0
d le t
a
b e t h e gre a t e r
.
c
h as
0
c
z
G
92
AL
H I H ER
m
E x a p le
Fin
.
mit
d
th e li
w wh h
b et
s
ic
een
2
ax
a
5x — 7 x 1
e c ap ab le o
v u
f all
al
ax
Pu t
h
t
(
‘
i
s
e
d h
in
o r er
d h
o r er
t
v u
th e
at
al
exp re s s io n
es o
49 1 — y
4
t at s,
2
2a 2 +
en c e
p o s iti
e
an
re al
y
q
u
tity
an
.
(1
49
2a +
is ,
2
fro
n
5y
a
2 2a
2 0a
4
2
2
at
y
4
fo
x
2
5
1 y
?
ratic
a
2 0a
n ega ti e o r z ero
an
,
real,
th e
e,
b e p o s ti
st
be
ay
b e p o s iti
st
ay
49
be
st
t is q
e
d 49
2o u
mu t b
s
e
.
No
t
a
2
f
at
a
u d mh u d m
u
v
(
)(
(
)
)m
h i ( 9 0)
u
v
m
i
(
(
)
)
h v(
u
v
) m
(
v
d
w(
)
( 9
v
)
(
)
(
i
h
d
v
(
(
)
(
)
h
v
w
u hv u
v
wh
m v u
v wh
m hv
md v u
In
t
7x + 5
— 7x
a
mu t li
a
'
3
5x
en
- -
b ein g
es , x
.
7x + 5
-
2
my b
G EB R A
acco r
a
2
2 0a
25
l 0a
i n g as
4
2
x
n egati e o r z ero
is
2
a
2
12
a
a
2 is
a
in g a s
n egati e o r z e ro
2 4 is
+ l oa
acc o r
,
n e gat
e o r z ero
;
.
1 2 , an d fo r
T is e x p res s io n is n egati e a s lo n g as a lies b e t e en 2 an d
2 0a i s p o s iti e ; th e e x p re s s io n is z ero
e n a = 5,
1 2, o r 2 ,
s c
al e s 4 9
en a = 5
H en c e th e li itin g al es are 2 an d
b u t 4 9 — 2 0a i s n egati e
1 2 , an d a
ay
al e
a e an y in te r e iate
.
.
E XA M PLE S
IX b
.
th e
m
mit
D e te r in e t h e li
e quatio n
1
.
b et
s
w
ee n
.
wh i h n mu
a
a
2
4
S
.
5
.
.
t
e
ts
at
z =
c
d th
o r er
at
0
.
real,
hw h
I fx b e
.
5 an d 9
6
Ifx b e
.
3
e real ro o
2)
lie in
st
c
—
my h v
.
p ro
x
x
real,
2
v h
e
1
x
2
1
x
r
o
p
t
v h
e
t
mu t li
at
s
h es b e t
at
x
2
x
w
e en
+ 3 43
2
“
2x
3
an
71
7
e
1
d
3
can
b et
w
e en
a e n o
an
d
v lu f
v lu b w
fo r all real
hv
1
a
a
es O
e
et
x
.
ee n
.
Fi
n
d
th e
eq
I f a , B are
2
1
( )
Q
( 2)
(0
u
at io n
ro o ts o
(a fi
z
—I
h
w
f th e
Ja
o s e ro o ts are
eq
+3
2
u
Ja
a tio n x 2
e —1
(B a
a
px
)
,
a
b
+ q = 0, fi n d t h e
v lu
a
e o
f
7
.
I f th e
8
.
If
e xc ep t s
t
h
uh
.
o se
th e
e xp re s s io n
w
I f th e
o f th e e q
9
u
2a
th e
eq
a tio n
b
S
.
v lu wh
es
a
11
.
12
.
h w th
e
d h
Sh w h
Fin
t
v lu
h as
n o
th e
I f th e
ro o ts o f
13
a
b et
es
ro o
.
(
+ c)
a
es
*
t
e
.
a
e, a n
o ss
en
122
x
d
s rea l,
s
(
h
ii —
b
een
ax
2
h
e o re
m
s an
ra se o
a
d
e x p res s io n
if a
2
n
ee
N
s
N
W)
e
,
.
n o
a
an
+2
hu
T
?
et
e
a
1
een
o r rea
6
2 v + 3x
ca
e
a
2
b
—
a
f all
e
o
an
d7
es o
.
v
.
.
— c ”
)
.
0 b e p o s s ib le
c=
o
)
(
2
b
ac
2
)(
an
2
30 +
e
a
)
d)
b ) ( dx
(
(bx
ax
an
d
0
2
s c
e
a
) (ox
d
2
d diffe re n t , t h e n
1)
will b
e cap ab le o
e sa
e s
c
e
c
c a l re a din
e
t
er
e co n
on
a
a
a
a ll
f
en
o
e re
en
n
re
en
s
.
sce
e
so
n
an eo
ro
uu y
s
ce
en
g
.
wh h i v lv
d wh
f
fu t i
ll d
ymb l f h f mf (
.
2( )
will b p b l
y v lu b w
f
l v lu
f
4x
ic
D E F I I TI O
A n y e x p res s io n
al e
is de p en de n t o n t h at o f x , is
n c t io n s o f x a re
s all
de n o t e d b y s
v u
Fu
4
h v h m ig
lu d t h i h p t
w it h m mi ll u
I t w ill b
v i t h t i t du
wh i h th t u d t w ill f qu tly
ti
b
2
l co n c
l
es
p
an
o
3x
.
al
ex a
x
z
e x p res s io n
c
2 bx +
m
h
y
l
t
d
p
g
m t wit h i h i mth mti
t
.
2
x
+ 2 bx +
2
es
a
v h
3x
b e) ( 2x
d
v lu
all
f
AC
p
th e
an
o
vice vers d
th e
at
We
w
(
x
ts
at
t
e
9
p
t at p h as
real,
d mi
v h
’
2
B
e o
q
p ro
,
+ 2 b x + c = 0 b e a an d B, an d
b e a + 8 an d B+ 8, p ro e t at
ax
A
a
fp
93
.
.
2
e
ra t io o
m
a tio n
vid d h
v lu f
Of a x
ts
will b imp ib l
Sh w h
14
v lu wh i
e
r
o
p
if x is
at
0 b e in t h e
e x p re s s io n
re at es t
g
e
t
e
re al
real,
is
en x
th e
at
u
U A TI O N S
EQ
a
d2
ac
a
10
an
2
=
+n
+ ux
ee n
Of
ts
ro o
Q u A D R A TI c
f lx
o
re a l,
OF
2
lie b e t
as
c
ts
ro o
be
x
Y
TH E O R
TH E
n
ca
es
o
x,
nc
a
e
t
o s O
e
o se
an
on
o
o r
x
x
)
.
,
96
i
u
t
q
t m t h t
u
h
t
q
f
my b
md i
u iv l
w ill p
y
( )
t o a s ta e e n
ro
t a a n y c a n ge
a e
n th e
al e Of x
d ce a c o n se
en
h
c an
e in y , a n d vi c e vers a
T
e q a n t it ie s x
g
an d
a re c a lle d v ari ab le , a n d a re
as th e
i
s
d
i
i
n
e
r
t
r
d
s
t
e
y
g
in dep en den t variab le a n d t h e dep en den t v ariab le
u
s
th e
e
a
on
h
s
x
a
e
c o n s ide re d a s
fu h
v u
.
.
eq
u
u h
a en
t
94
HI
GH E R
AL
GE BR A
.
y wh h m h v
v u
u
ic
is a q a n t it
a
a e an
y
y
a l e w e c o o s e t o a s s ign t o it , an d t h e c o rre s p o n din g dep en d en t
in e d a s s o o n a s t h e a l e o f t h e in de
al e de t e r
va ria b le h a s it s
n
a ria b le is k n o
e n de n t
p
A
in dep en den t
n
v u
h
v u
v
A
n
m
w
.
e x p re ss io n
x
p0
where
va ria b le
"
+ p ,x
o
'
H
f m
f th e
+ p 2x
or
—2
n
fi
is a p o s itive i n teger, a n d t h e c o e f c ie n t s p o , p , , p ,
d
o
p
n o t in
o l e x , is c alle d a rat io n al an d in t egral algeb raic al f n c t io n
ofx
I n th e p re s e n t c ap te r we s all c o n n e o u r a t t en tio n t o
n c t io n s o f t is k in d
fu
n
v v
.
h
A fu
h
.
h
u
fi
.
.
w
hu
is s a id t o b e lin ear
h e n it c o n t a in s n o
s ax
b is a lin e a r
h igh e r p o e r o f t h e a riab le t a n t h e rs t t
is s a id t o b e q a drat ic
n c t io n
it
A
en
of x
n c t io n
ig e r p o e r o f t h e a riab le t an t h e s e c o n d ; t
s
c o n t a in s n o
”
n c t io n o f x
n c t io n s o f t h e th ird,
ax
b x + c is a q a dra t ic
ig e s t p o e r o f t h e
th e
de gree a re t o s e in w ic
t h e th ird,
T s in t h e la s t
a ria b le is re s p e c t i e l
th
n c t io n o f x o f t h e n
de gree
a rt ic le t h e e x p re s s io n is a
*
1 24
.
w
n c t io n
v
h
fi
fu
fu
h h w
v
u
fu
s
h
h h
v
v y
fu
1 25
Th
ymb lf ( y ) i u d t
d
i bl
d y th u
by
i
t
i
v
l
y
li
u
d
fu
d
p
q
*
e s
.
va r a
es x an
ec
n e a r an
Th e
ra t ic ,
ra t ic ,
equ a t o n s
e
f
0
x y
i
x
)
( )
(
a s th e
i
a c c o rdin
u
n
t
o
s
n
c
f
g
.
:
-
h
Fu
h h
.
0
w
hu
de n o t e
fu
i
g
o g + dx
a
n ct o n
b xy
n c t io n s Of x ,
y
2
ax +
o
f( ) f
,
f t wo
e
y
+
f
.
id t o b e lin e a r, q
a re lin e a r,
x,
q
( y)
a re s a
x
hu
.
o
ra t c
a
f
se
c, an
s ax
a re re s
.
s
x,
o
wh
u
.
u
u
d
ad
a
.
*
126
bx +
°
ax
wh
hv
dm
it s
We
.
a
c
e re a a n
hus
s v
u
d
B
a
a re
v
h
rt
A
1
2
at
h
t
t
0
e
ex
r
ss
i
e
on
p
p
i
o f b e in
t
n
t
h
u
e
or
a x
a
x
,
g p
(
B
)
)
(
”
t h e ro o t s o f t h e equ a tio n a x
bx c _ 0
e
ro
ed
in
.
f m
.
u
a
a dra t ic e x re s s io n
T
x + bx +
a
p
q
re o l e d in t o t w o ra t io n a l
ac to rs o f t h e
2
t h e e q at io n a x + b x + c
0 h a s ra tio n al
2
b
4 a o is a p e r e c t s q a re
2
f
f
u
is
ca
fi
*
.
th e
f(
fu
x,
i
by
n ct o n
y)
:
ax
o
f b e in g
wh v
h
h
fu n ctio n of x
.
D en o te
le
er
en
.
1 27
To fi n d th e co n di tio n th a t a qu a dra tic
a
e reso lved i n to two li n ea r a c to rs
b
y
f
m
ab
p
rs t de re e ,
n e
e
g
ro o t s ;
t a t is , w
c
2
f(
x,
+ 2hx y
h
w
y)
by
2
e re
2gx + 2fy
c.
,
y
HIGHER
96
Fo r
.
wh
G EB R A
E XA M PLE S
*
1
AL
u
at val e s o
f
mwill th
IX
.
2
eq
n
u iv l
a en
3
d
Fi
.
t to
S
.
u
v lu
du
p
th e
th e
h w th
e
a
es o
m
y
—3
ro
th e
at
w y d mi
4
hv
a
ts
s a
a
o
I f th e
.
e a co
f two
eq
u
o n ro o
d
n
th e
real
t,
s
(
x
2
-
2
y
)
— r
y
a
6
v
a
e a co
t
e
t
s
be
ts
ro o
o
7
.
my b
a
8
.
res o
fc
lin ear
on
f th e
e
n
ax
t
.
If x
x
en
2y
t
a
an
dy
et
t
n
v u
e
th e
at
by
x
2 ax
p ro
n
'2
y
2
e o
x
a real
are
t
Zy
3
h
a
e re
e a ra t o n a
2v
e
.
is a
o fx
t
I
at
mu t
’
'
2
one
o
f th e
'2
2 /i xy
by
my m m
y+
f th e fo r
x,
3y
x
.
re al
3 5= O,
v lu
a
e
o
f y,
qu
2
d 6,
an
t it ies
9 2x
an
c o n n e c te
o
an
d by
d fo r e
v
th e
u
e ry re al
fy
.
eq
atio n
2 oy + 2 4 4 = O,
d y b et
w
ee n
1
an
d 10
fi n d th e
n c t io n
be
s
.
real
an
2
.
Q ay + y
e en
e
a c to rs o
‘
v lu
t wo
v h
1
at o n
e e
3 xy
f
a x
,
4y
e x p res s io n s
’
2
e
.
a
2
ac to rs ,
t
on
s
h will li b w
If
10
i
l fu
my b
t
lx
,
v d ivi ib l b y f
h
i
u
Sh w h t i
q
h
my
’
.
Q iix y
2
e res pe c t i ely
v
P
"
ac to r
n ear
e co n
e
’
0
''2
at o n
fo r e e ry real al
val e o f y t ere is
9
f
y
er
e x p re s s io n s
z
2n
2
to
x
u
th e
h
e x p res s io n
e
Fin
P
at
n
9
l
9
v d i li f
u
i
q
d th d i i h
l
0)
e e it
a
on
my
3x
c an
.
.
mu t b
it
at
d i i th
co n
mm
I f th e
.
B
(
.
lin e ar fac to rs
hw h
2
la:
my h
e
at io n s
mm
Fi
a
e x p res s io n
9
.
mk
f t wo lin ear fac t o rs
1) d
5
fac to rs ?
will
c
ct o
A
al
ratio n al
mwh i h
f
.
e ex p re s s io n
2
x
2
x +
+
+
y
y
f res o l tio n in to two
a b le o
p
ca
c
.
2
be
.
.
co n
di i
t
on
t
h
at x
C H A PTE R
MI C ELLA N EO U
S
1 29
c e lla n e o u s
v
u
q
e
h
t is
In
.
c
a t io n
s
h
a
w
t
r
e
e
p
l
e
m
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4
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c
x
2
mpl
in E xa
re s p e c ti ely an
v
2
.
zx
2
b z
y
f th e gi
an y o n e o
19
h
2
y
“
2=
h
multipli
cro s s
2 2
b c
u
2
x,
,
.
x re s p e c ti ely an
,
c x
x
S b s tit te in
z
y
,
13
dt
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Fro
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=
dt
an
an
z
E
2
z
atio n s as
11
2
U A TI O N S
EQ
3z
u u
2,
4
:
US
we o b tain
atio n
x
Fro
EO
Ja
.
X
.
c
.
e
b
e
0
6
2
3a b
2 2
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o
b tain
11 0
AL
H I GH E R
x
2
y
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z u
1 2,
x
2
y
2
2
zu
8,
x
2
yz
2
G E BR A
2
u
2z 2 2
u
3 xy
1
12
.
3y z + y — 6 z
52 ,
=
7 z , y z + x = 8z ,
=
+y
.
2 xy
.
xz
4
x
2
2
2
+y + z
a
2
,
3x
2
—
— 2z = 3 a
=
z + zx
x
6
a , 3x +
y
y
y
I
1 38
t io n
E TE R M I
n
o
o
E Q U A TI O
A TE
f ll w i
u
S ppo s e th e
.
N
g p
b le
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N
S
.
p ro p o s e d fo r
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l
:
A
h
ND
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o rs e
4
1
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n b
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c o s t s £ 2 3 an d e a c
L et
b
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en
s
rso n
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th e
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46 1
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al e w e p le a s e t o x , w e c a n
a n d it is c le a r t a t b y a s c rib in g an y
al e fo r y
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t h e n at re o f t h e q e s t io n t a t x an d y
B u t it is c le a r ro
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it t is re s t ric t io n , a s we s all s ee
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XX
I
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112
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HI
GH E R
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at
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in teger
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m
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h
1 419
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v
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T is is
an y p o s iti
an d y ; t
u
v
th e gen e ra l s o lu ti o n o f th e e q atio n . an d b y gi in g t o p
in te gral al e o r z ero , we o b t a in p o s iti e in tegral al e s o f x
c alle
e
s
e
a
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p
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=
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2,
v u
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1 7 , 28, 39,
th e
n
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um
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xa
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3
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um
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d
h
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d
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v
h whv
So l tio n s are Ob tain e b y as c rib in g to p th e al es 1 , 2 , 3 ,
9 ; an d
b er o f ay s is 9
I f, o e er, th e s u
ay b e p ai
t ere fo re th e n
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ay als o
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a e th e
al e s 0 an d 1 0
If p z o ,
in
is p ai en tirely in fi o rin s ; if p = 1 0, t en y = 0 an d
t en x = 0, an d th e s u
i s p ai e n tirely in alf c ro n s
th e s u
T s if z ero al es o f x an d y are
b er Of ay s is 1 1
i s s ib le th e n
a
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-
.
.
-
.
.
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ere
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.
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f eac ?
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x
E li
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e ac
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a
mi
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l 0x
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Th e gen eral
,
we o b tain
u
so l
tio n
o
h
f t is
43
5g
2z = 2 2 9
.
8x
33;
1 43
.
u
eq
z
y
ati o n
is
,
an
d
c
il
ren
,
resp ec ti ely
I
H
e n ce
by
s
u
ND
E TE R M I
u
v
m
hu
.
A TE
U A TI ON S
EQ
113
.
we o b tain
b s tit tin g in
H ere p c an n o t b e n egati
T
s
fro
1 to 5
N
5p
z
e o r z ero
x
:
3
b ut
,
my
.
hv
a
4,
v u
in tegral
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es
5,
1,
2,
4,
7 , 1 0, 1 3 , 1 6
3,
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p o s iti e
e
a
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5,
z
7 , 1 2, 1 7 , 22
2
:
E XA M PLE S
v
So l
1
.
4
.
1 3 x + 1 1y
d th
=
4 14
.
:
5x + 2y
.
5
.
.
53
=
2 3 x + 2 5y
.
u
.
9 15
=
.
v
ge n e ral s o l t io n in p o s iti e in tege rs ,
an d
w
i
c
s atis fy th e e q atio n s :
y
n
fx
in tegers
e
2
Fi
o
v
in p o s iti
e
X d
.
7
e
h h
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3
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8
.
10
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13
A
11
.
.
.
£3 7
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h
.
6
.
9
.
12
.
7 x + 12y
8 x — 2 1y
7 7y
—
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=
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at o n e
a
s
be
an
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29 5
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valu e s
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I n h o w an y ay s
14
in c l in g z ero s o l tio n s 2
.
.
s £ 7 52 in b
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3
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.
Wh at
mpl
way fo r a p ers o n wh o h as
an o t e r wh o h a s o n l
ro
a
lf
c
ns
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is th e
si
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.
ly gu in e as
56
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17
mi d 7 H w m y u h u mb
h ?
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18
di h g d b f £ 6 d if h h g i b p id i h lf w
.
t
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2
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an
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.
.
,
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e c
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or ns
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t
39
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C H A PTE R
PE R M U TA TI ON S
139
so
m
.
e o r
E ac
so
m
e or
hu
T
le t t e rs
o
um
h
o
um
m
a,
a,
b,
h
ft
h
o
c a lle
co
mb i
ft
h
c a lle
a co
ca ,
da ,
c
b,
db ,
dc ;
wh i h
m
e a re s ix
ti
re s e n
c an
c
a c,
t in g
a
in
a rra n
md
u mb
a
n
er :
diffe re n t
a
.
e
a
th e
n
e
,
d,
mt
e
g
be
b c,
d,
e
e r, n a
b a,
i
b y t a kin g
e
n at io n
n
c
diffe re n t
.
v
b d,
a
mb i
a
b c,
t in g
md
a
k
b y t a in g
e
a
md b y t ki g
mly
umb
be
t we l e in
e a re
md
u at io n
be
c an
d
.
mt
p er
a
a d,
n at o n s
p
c
be
c an
a c,
at a
ese
d
wh i h
c an
c
m
ab,
eac
c
b,
re s en
p
es e
d t wo
c,
h
wh i h
t h e p er u ta tio n s
b , c, d t wo a t a t i
s
Th e
whi h
f t h e gro u p s o r s electio n s
b e r o f t in gs is
a ll o f a n
o
a
eac
m
h
.
C OM B I N A TI ON S
AND
f t h e a rra n ge en ts
n
b e r o f t in gs is
E A CH
a ll o f a
h
XI
en
o
f t wo le tt e rs
.
b y t a kin g t h e le t t e rs
n a
el ,
my
b d,
ed
l
s e ec ti o n
o
f t wo le tt e rs
.
f mi g mb i i w
ly
u mb f t h i g h l ti
t i
d wi h h
;
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w h v
i
l
id
wh
i f m
t
th
h
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m
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i
ft
t
t
d
;
p
g
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b
d w mk
if f mf u l t
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i
h
thi
d
t
b
g
f ll wi g w y
F mt h i
s
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or
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s
e s ri e
b c,
to
a cb ,
s ix
b ca ,
b a c,
ca b ,
mu t
diffe re n t p e r
cba,
a t io n s
.
on
co n
co n s
en
e ec
a re o n
e
on
o
s
a
an
e
t e rs
s n
a
n
o
a rs
e
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t
t
n
r
s
c,
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or
n
e re a s
it
a n s
er
e
o r n s an c e,
o
a rra n
ree , s
e
n
c
e
116
HI
GH E R
AL
G E BR A
.
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1 43
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1 20
HI
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mb
146
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co n
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m
n
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147
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15
regi
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1 5 recr its
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ps
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PE R M
U TA TI O N S
C O M B I N A TI O N S
AND
121
.
m
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14 8
I n t h e e x a p les w ic
it is i p o rt a n t t o n o tic e
o llo
la fo r p er u ta tio n s S o ld n o t b e
t at t h e o r
s ed
n t il t h e
s it ab le s elec ti o n s re
i
re d b
h
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de
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q
.
.
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e n an d 4 A
eric an s
7 E n glis
1
an y
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o n e,
2 at le as t 2 A eric an s
erican s ,
ro
.
b e fo r e ; in h o
tain s e x ac tly 2 A
1
Th e
e
n
th e n
b er
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We
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.
b er o f ay s in
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e n c an b e c o s e n is 7 0 4
it e ac o f th e s ec o n ;
en c e
p s can b e as s o c iate
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4
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s
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1 22
m
m
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ts in th e 4
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re
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sec o n
th e req ire
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b er o f
n
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h
f th e letters
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a e
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1 1 , fin d
2 25
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be
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14 9
a e b e e n re a rde d a s u n like
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in s t an c e , in E x 2 , A rt 1 4 8 , t h e c o n s o n an t s a n d t h e o wels
a
y
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on
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e kin d ;
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in di id a lit
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fi d ll t h p
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1 50
Supp s w h v
f
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i g 12 b
k
h lf 5 f t h mb i g L ti 4 E gli h d
h
mi d i diff t l gu g
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h l g g my b g d d b l gi g t
k i
l
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PE R M
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A TI O
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1 25
.
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t h e s a e la n g a ge a re n o t dis
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ld a e t o fi n d t h e n
o t e r, w e S o
t in gu is h a b le ro
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o f wa s in
a re e x ac t l
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a lik e o f o n e
in d, a n d 4 e x a c t l a li e
ic is n o t dire c t l in c l de d in a n y
o f a s e c o n d kin d : a p ro b le
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s elves , ta kin g th e
a rra n ged a
p
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qf
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th e res t a ll difieren t
1 51
To
n
m
.
m
m
m
m
m
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h
Le t t
b e b, r
to
be
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t o b e e, an d t h e re s t t o b e
um
u
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u
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n lik e
o
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m
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h
f mh
m
h h
b er o f per
t at io n s ; t e n if in
b e t h e re q ire d n
t
a t io n s t h e p le t t e rs a we re re la c e d b
an
n e o f t h e e p er
o
p
y
y p
an
o f t h e re s t ,
ro
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di fe ren t ro
t
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s
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w md i h f th p mut ti s
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.
p
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lik e le t t e rs ,
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n
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r =
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t h a t is
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h
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b y re p la c in g t h e
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ll di fe ren t
my
a
be
1 26
H I
m
E x a p le 1
letters o f th e
w d
hv h
um
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H
d
an
n
GE B R A
mu
k
wh h
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1 3 letters o f
tatio n s
b e r o f p er
mu
ere
a e
th e
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or
my
AL
.
tatio n s
iffere n t p er
ass a ss in a ti o n ta en all t o get er ?
H ow
.
GH E R
|
H
4
are 3 ,
h
c an
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be
2
md
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.
13
E
W
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1 001
m
um
m
md
n
b ers c an b e fo r e
an y
H ow
E x a p le 2
1 , 2 , 3 , 4 , 3 , 2 , 1 , s o t at th e Odd digits al ay s o c c py th e
Th e
h
.
d
dd igits 1 , 3 , 3 , 1
o
be
can
arran
w
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d h u
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igits
p lac e s in
I
2 2
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4
v d
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e en
igits 2 , 4 , 2
c an
be
I
E ac
H
h
th e
en ce
1 52
m
tim
ti
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w
f th e
req
ire
fi
e,
wh en
es
in
H
e re
n
y
th e
a rra n
we h a v e
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y
a rr
an
fi
h
fi
t
m
g
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en
be
L
as s o c iate
6
x
eir
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co n s
it
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e ac
o
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in
ay s
.
ons
o n c e,
ea e
er
e
en
n
s
e
e n
er o
n
w
wh
h
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p
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my b fi ll d p i
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i
p
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o
11
th in gs
twi c e,
u
r at a
p
to
r
.
u
n
ree
dw h h
3 : 18
x
er
er o
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ac e s can
ll
b
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p
dis p o al, e a c o f t h e
s
h h
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p
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To
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um
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2
t wo pla c es c an b e
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ace
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a n n e r, a n d n o t ic in
h
at at an
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in dex o f n is alwa s t h e s a e a s t h e n
b e r o f pla c e s
lle d u p,
w e s all a e t h e n
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um
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fi
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y
um
wh h
fi
s
1 28
th e
v lu
es
+
1
a
HI
GH E R
in
u
1 , 2, 3
1 b ec o
s
m
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r
u
a
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c c e s sio n
l to 1
.
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,
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r
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u lly i
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n
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.
f mu l f th u mb f mb i ti f h i g
t
f u d w i h u t u mi g h f mu l f h
t im my b
u mb f p mu t ti
L t C d
h
u mb f mb i i f h i g t k
t
h
l t
t im
d l t h
thi g b
d
t d by
1 55
n
mi
m; t h
m+
l
B, we fi n
"
97, —
+
taken
s
a re
g
in
f r
1 th e
o
0m+ 1
th e
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f
o
e
0
es
”
l to 2
m
t
o
p
2
an
a
2
r u
r
n
n a t io n s
a
.
2
d fo r
v lu
t h e gre a t e s t
o o se
u +
an
.
at
b,
er
en o
r
e
a
c,
a
”
d,
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a
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o
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on s
or
an
e
t
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o
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ass
t
e
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or
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t
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.
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e
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e
n
t
s
n
t
e
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en
e t e rs
U TA TI O N S
PE R M
C O M BI
AND
N
N
A TI O
S
.
m
h w h
f m
m
m
h
h w
hu
h
m
m
h
um
wh h
f
m y
um
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i
h
b i
h
i
l
t
t
Th
f
x
u
u
m
b
f
mb i ti
q
i
t g th
w ith t h t h t t i b
t im wh i h
h th t t i
d
i g th
mb i ti i th i m
hp
B t by f m
t
t
i
m
F
i l
will b
d
i
t
if
3 th
p
mb i ti
f u d m g th
b w ill b
t i i g
mg
d m g th s
t i i g
h
t i i gb
H
eac
o
n
a
s
k
y
wa a ; t e n
it t h e re a in in g le t te rs we c a n o r
b in a t io n s o f n
1 le tt e rs t a k e n a
l at a ti e
co
W it h
t
s we s ee t at o f th e c o
rit e a ;
o f t es e
b in a t io n s
b e r o f t h o se
ic
t in gs 7 a t a t i e , t h e n
c o n t a in
b er o f th o se
th e n
ic
c o n t a in
C —l ’ s i ila rl
C” , a n d s o fo r e a c o f t h e n le tt e rs
Ta
0
e
a
.
r
s
.
e re o re
r at a
t
o se
co n
a
co n
a n
0, an
o r
n
t
on
n a
a
a n
o se co n
n
,
on
on s
r
es
n
By
w iti
r
mil
Si
n
y
l
ar
g
1
n
“
d
fi
y
n all
u
eac
ro
a
co n
on s
a n
11
—2
”
3
2
an n er e ac
,
n s a n ce,
o se co n
,
a n
n
a n
0
r
n
ar
:
e
,
a, a
on
en c e
.
x
— l in s t e a d
—r + 2
C2
’
n
—r
0
o
fn
an
d
'
9
re s
p
e c ti
v ly
e
,
X
_
r
2
,
yt
o
s ide
”
h
g
t
h
u
;
et
O'
er
n
v
th e
+l
x
C,
— fr + 1
= n
M lt ipl
f m h
9
o se
na
,
n
an
d
an
co
s
or
.
e co n
o
0
'
n
on
a
on
a
an
er o
er
e
n a
o
e n
.
co
e
c
n
o
ea e
re
o
a
a,
so
e
o
e
c u ar o n e
co
ta
c
e
u
t
s e
n
e rt ic a l c o
.
um
l
n s an
d
c an c el
lik e
f
a c t o rs
s
(
n
— l
7
'
(
)(
r
n
— 2
— 1
)(
)
r
—2
)
m
1 56
To fi n d th e to ta l n u b er of wa y s in wh ich i t is p o s s ib le
a ke a s el c ti o n
to
+ r
b y ta kin g s o e o r a ll o u t of p
q
th in gs , whereof p a re a like of o n e kin d,
a like of a s ec o n d kin d, r
q
a like o
f a third ki n d ; a n d s o o n
m
e
h
Th e p t in gs
t a e 0, l , 2 , 3 ,
dis po s e d o f in q
k
so
on
.
H
m
.
.
H
.
A
.
my b
a
.
e
+
o
h
y
m
m
p
l w y ; h
h
o
f t
a
s
e
t
.
e
wy
m
m
a
a s ; fo r w e
f in p + l
y
b
e
a
Si ila rl t h e q t in gs
y
wa y s ; a n d
7
t in gs in
dis po s e d
°
h
H I GH E R
H
dis p o s e d
ta
w
k
a
th e
en c e
f is
o
n
(
u mb
1 ) (q
19 +
h
u
f
j
th
;
s
is
y
(
r
g
.
wh i h
in
l l)
ys
c ase
re e c t in
e re o re ,
a
l)
l
-
B u t t is in c l de s t h e
en
w
f
er o
G
A L E BR A
a ll
c
my
t h e t h in gs
be
a
-
in
t is
h
wh i h
o
th e
c a s e,
h s
um
f t h e t in g
t o t al n
b er
n on e
c
a re
f
o
l)
1 57
A
f mu l
e n e ra l
or
e x p re s s in
a
h
t
e
g
n
u mb
er o
h
m
m
u
m
E
pl
g m t
ran
en
e
o
,
t
i
o
n
r
o
r
o
p
p
h
Fin
e.
xa
f fo
d
u
th e
e re are
1 0 letters
T
re e ali
2
T
o ali e ,
3
T
o ali e ,
4
A ll fo
t, i, n
3
3 p airs ,
4
ree
e se
T
s
w d
or
th e
o
t
er
c
as
en
be
ifferen t
o
gi
v
r, r.
p p
c an
th e
h
h k
hv h
be
s ix
er o
re
0,
to 5
[
to
1
or
h v
w h v
w
k d
h v
h i
w h v
mu
u
ay s , as
e
T is gi
.
is 5
on s
20
7
.
to ta e 4 iffe ren t
1 5 s electio n s
a e
es
3 + 30 + 1 5
e
.
t
at
a e
s,
53
.
to per
te in
.
arran
ge
mt
en
s
.
mt
18
arran
ge
3 60
arran
ge
or
2 2
o ut
.
04
t, i , n
r,
p,
I
3 x
two
w
x
6
in
e
f s elec t
E
o o se
in 3 1 0 ay s ; fo r we s elec t o n e o f th e
ain in g 5 le tters
T is gi e s 3 0 s elec tio n s
e
a
a
c
.
.
4
e s ri s e
en
s
.
L
.
hu
T
s
1
4
v
v
3
( )
gi
es ris e
to 3 0
x3
4
( )
gi
es ris e
to 1 5
x ]4
th e to tal
n
um
.
s
in 0 2 ay s ; fo r we a e to
T is gi es 3 s elec tio n s
4
es ris e
w
n
.
w
u
wv
3
a e
,
th e to tal n
gi
fo llo
t; i,
a e in 5
be
ay s ; fo r eac
o f t h e fi ve le tte rs ,
it th e s in gle gro p o f th e t ree li e le tters 0
c an
s s elec tio n
fro
d
r;
.
p airs 0 , o
o o se
t
b e clas s ifi e
0,
.
'
(2 )
ar
.
ers ali e
t
o o
t
e
I n fi n n g th e iffere n t arran ge en ts o f 4 le tters
o f th e fo rego in g gro p s
ay s eac
all p o s s ib le
1
a
.
m
s o rts , n a
ay
eren
T is s elec tio n can b e
th e
an d t en t o fro
le tters to
T
t
s elec tio n c an
Th e
f he t
b ut
fferen t
r
b e ta
c an
,
t
r
e, o n e
s elec tio n
Th e
2
th e
en
1 a s elec tio n , ( 2 ) an
th e le tte rs o f th e
mly
ifferen t
f s ix
o
f fo
o
1
1
o
md
a
.
I n fi n in g gro p s
p
w
m
m wh
f w m
wh h ( )
ic
b er o f ay s in
be
c an
a e fro
letters
r
v
um
n
mwh
m
d
d
u
u h m
h
k
d
if
f
( )
k
w
h
k
w
( )
k
h
w
d
w
( )
u di
( )
md
( )
k wh
m
d
( )
h
h
t
m
h
dm
( )
h w m
m
h
d
i
( )
h
m
i
hu
umb
m
di
d
w
h
L
v
x
( )
T
,
f
h
mu t
f pe r
g
t io n s , o r c o b in a t io n s , o f n t in gs t ak e n r a t a t i e ,
a
so
at c o
b
e
e
li
c ate d
t in gs a re n o t a ll di fe re n t ,
y
p
i
n
a
b
e s o l e d in t h e
o llo
an n er
a rt ic la r c a s e
y
g
p
.
b er
o
or
or
f arran ge
3 60
mt
en
arran
s
ge
mt
en
m
en
s.
ts
.
h
is 2 0 + 1 8 + 3 6 0 + 3 60 ; t
at
i s , 7 58
.
HIGHER
1 32
17
m
A telegrap h
.
s ign als
h
t
h as 5 ar s an d
in g t h e p o s itio n o f
be a e?
ud
in c l
t
i
i
o n s,
o
s
p
.
h mi
wh i
s
ar
e ac
re s t
md
wm y w
at c an
G EBR A
AL
at
;
m
a rin
di
s t in c t
f 4
t o t al n
th e
s
an
ay s c an 7 p e rs o n s fo r
In h o
o
eric an s s it
en an d 7 A
ay s c an 7 E n glis
t wo A e ric an s b ein g t o get er ?
18
c ap ab le
g?
o
umb
f
er o
m
I n h o w an y
ro n
t ab le , n o
hm
dw
ud
m
h
m
ib l
d w
mfm f m
I h wm y w y i i p
19
v ig h lf v ig
w fl i hilli g
i i g
b g
i
f
h
d
?
g
p
y
l
w
l
d
h
m
u
20 F m
g
y
pp
f
ki g l
h ki d ?
b md
i
ff i
Fi d h u mb f d iff t w y f d ivid i g m h i g i
21
l
u
u
g
q
p
b md b y h i i g fl g
f diff
H w m y ig l
22
h
h wh y umb f h mmy b h i d
b v
l u
fl g?
? H w m y wi h
Fi d th u mb f p mu i wh i h
b f md
f
23
i t k
h g h ?
h l
f h w d
i
l
i
h f whi h i h m
Th
24
p
p p
wi h h
i
f
wh
i
h
l
l
i
h
h
igh li
m
i
p
g
g
u mb ( ) f igh li ( ) f i gl whi h ul
li
fi d th
f mj i i g h m
i
f u
f wh i h
i
h
m
25 Th
1 p i
p
i
h
i
ll
h
m
l
w
h
wh
h
l
i
f
i
fi d
p
p
p
g
h
h tiigh fh pi
h w m y pl
diff
i
b k
h
Th
d 3
f
fi
h
d
26
p
th m
u mb f w y i whi h l i
b md f m
F i d th u mb f l i
d f
m
h
t
b
27
g
f mh w d p i
md b y ki g l
H wm y p m
u i f l
b md
f h
28
f h w d
mi i
l
29
F i d th m f ll u m
b
t
t
h
f
d
b
0
m
g
y
h
d
i
i
d i y u mb
digi b i g p
7
g
g
30
Fi d h m f ll u mb g
th
0 00 f md b y
h
digi
d igi b i g p d i y u mb
6
g
If fp
i
31
lik
h
d g b
lik
d h
q
t
g p b
dfi
h w h h l umb f mb i i i
Sh w t h t h
32
umb f p mu i whi h b f md
f m l
wh i h
i h
umb
b i g
wh
th
f
i
t
h
u
u
l
m
b
fb
q
If h
33
umb
b
l
b
ll diff
d
f
h
h m p imm u mb p v h h umb f diff f
f h
w
.
ta
co n
a
a
a
en n
t
t
s
a t o n ce
t
ne
t
t
t
t
.
t
u s 1n
o
s
t
e
a
2
e
2n
ro
o
’
a s
a
t
.
t
e
a
r
e x p re s s 1o n
on
n
t
en
ons
o
c
to
ree
et
t
an e , n o
an
n
n
4
a s
e
a
t
s
t
,
s e ec
t
n
s
to
n
o
e ren
e
o s te
e
ou
t
e sa
t
e
or
t
o
er
are a
2
n es ,
ac e , n o
c
n
c
are
t
e sa
tr an
o
n
e
e s tra
es
re s
t
t
e sa
e
c
e
s , an
or
e
ree o
9
co
a
t
a
,
o
9,
a
,
t
n o
8,
n o
n
s
to ta
t
n
o n
ts
en
n
;
.
t
n
;
e
e
.
ts t
a
can
e
.
e
e out o
a
re
t
’
t
e
er a s o r
er o
+1
n
ers a ,
ro
(m
e
eate
s
s
e
or
er
n
1 0
an
.
or
n an
e
er
n
.
t
e , an
n at o n s
t at
’
n an
e a
co
er
1 0 00
an
eat e
e , an
er o
er o
are e
is
an e
eac
o
c an
reater
e n
e a
e n
b cd
e
e
t
n
e
ro
e
ress o n
re
e n
e rs
n
e n
e r,
t
es
rea er
ers
n
t
o
e sa
arran
o
e tte rs
4
ons o
t
e
ex
t
n
a
n
are
c
n at o n
+r t
t
a n
s e e c t o n s an
t
r o
o
are
co n
oo
ro
o
c
e n
can
re e o
c
,
,
tat
ex a
at
e n
a
,
n
n
t
er o
n
a s e ec t o n c an
er
a
a
ro
a s
o
eac
o st n
tat
s tra
er o
e t t ers
s e
o n ey
or n
es ,
s o
— 1
.
n , a
o ran
e
an
er
e re n
c
+
o
t,
a c ro
o
s
e re are eac
t e su
ts 0, 2 , 4 ,
e
e re n
on
t
ts l , 3 , 5,
.
'
ts
e su
n
t
u s 1n
o n
ett ers
or
e
.
0
t
4
n
.
0
an
e
a
t
o
n
n
t
e tt e rs o
,
a su
.
s
o
.
n
a
e
n a
1
e n
ta
e
a
ts
a re n
n
.
ra
e as t o n e o
at
en
5
e ex c e
e ex c e
a
er o
n
e
ere
.
e re
so
e ren
ser es
er
an es
an
o
t
ere a re
an e
to
e
es , an
n
er o
o n
e n
t
-
a
o
t
or
e
t
o n n
,
a
er,
an
n e
.
t
e o
e n
n
;
ro
t
e re are
s tra
a
n a s can
s
e
n
.
1
o
ett e rs o
e
n
o ss
.
o
.
er
an
rs o n e a
co o
s
ts , 4
e, t a
a
e
e n
o
.
t
s
n
co co a n
c an
ro
a
n e
3
n
.
art
a
ru
on s o
e re
a so
n
ro
.
t
n
n
a
,
a
an
o
n
.
at a
n
e re s
s
.
ons
c
reat e s t
c an
en
e
or
e
e n
er
’
s
,
.
0, o
t at t e
”
1) 2
1
.
,
n
e a
er o
e re n
e re n
t
t,
an
e ac
o
\
ac to rs o
t
e
XI I
C H A PTE R
MA T
1 58
de
m
HEM T I
A
m
o n s t ra t e d
dire c
fi
n d it c o n
en tl
e n ie n
q
at h e
at ical in d ct io n ,
u y
m m
m
f th e fi rs t
We
a
v
.
C TION
.
a
o
o
a
e
e
o
or
ca
ro o
e
o
c
e s
e
a
s
n
ee a re n o
u
c
o
o
c as es
ro o
n n at
pp o s e
it is
b ers is
ral n
ire
re q
al
eq
s t ra e
n o
a
t o p ro
e
t
th e
at
su
t
y
e a s il
w e fre
fk w
wh i h w h ll w illu t
u d
v h
m
Su
u um
u
m
h
u m
m h w mh
j
u
um h
u wh
u
)
m i
mh
h d h
(
u
E x a p le 1
o
by
DU
mt h mti l f ml
t md
f; i u h
fp
mpl y mt h d f p
t t
M A N Y i p o rt an t
.
IN
C AL
.
n o
as
n
.
o
f th e
c
u
b es
to
uh
h
b y trial t at th e s tate en t is t r e in s i ple cases , s c
i g t b e le d t o co n ec tu re t at
t is
e
en n = 1 , o r 2 , o r 3 ; an d fro
as
la was tr e in all cas es
A ss
e t at it is tr e
en n ter s are
th e fo r
ta en ; t at is , s p p o s e
c an eas ily s ee
wh
mu
k h
to
A dd th e
m
.
(
n
+
ter
to
n
,
t
at
is ,
m
+ 1 ter
s
n
n
ter
s
to
+
wy }
h
(
u
(
n
+ 1)
-
Z
(
e ac
si e
; t
en
2
U
2
1
n
n
2
+
n
4n
+1
4)
4
u w umd
u w m
wh h k f
mf m
h
w
d
u
u
wh
k
um
whm w
m wh v h um m
h um
h
u wh
w
k h
u wh
m
k
h
u wh
k
hu
u i u u v
m
e
to b e tr e fo r n te r s ,
is o t h e s a e o r as th e res lt e as s
en
e ta e
o r s , if th e re s lt is tr e
n + 1 t a in g th e p lace o f n ; in o t er
en
e
b er
ay b e , i t i s t ru e
ate er t at n
a c e rtain n
b er o f ter s ,
e n 3 te r s are
e s e e t a t it i s tr e
in c rea s e t at n
b er b y o n e ; b u t
en
s are t a e n ; i t 1 8 t e re fo re tr e
en 4 te r
ta en ; t ere fo re it is tr e
n i e rs ally
s th e re s lt s t r e
T
5 ter s are ta en ; an d so o n
ic
.
.
HIGHER
1 34
m
pl
E
a:
a
xa
2
e
d
To
.
mi
e te r
AL
du
th e p ro
ne
G E BR A
.
f n b in o
ct o
m
mi l f
f th e fo r
ac to rs o
a
.
u mu
By
m
( +
(
13 + a
hv
ltiplicatio n we
ac t al
a
)(
)
x
a e
a + b)
(
+ b)
(
a
2
+
33
+
c
)
abc
+
x
;
x3
(a b + a c + a d + b c + b d + c d) x
ab cd
ac d
b
e d) a
:
b
a
b
d
a
c
+
(
2
'
.
h
In t
1
b in o
e se res
u
v h
lts we
o b s er e
um
d
m
Th e n
b er o f ter
ial fac to rs o n th e le ft
m
.
t
s
at
th e fo ll o
h
rig
th e
on
.
w
wh d
m h
in g la
t is
m
on e
ol
s
o re
t
:
an
th e
n
m
um
um
b er o f
Th e in e x o f a: in th e fi rs t ter
is th e s a e as th e n
b er o f
b in o ial fac to rs ; an d in eac o f th e o t er ter s th e in ex is o n e le s s t an
t at o f th e p recedi n g ter
2
m
h
.
h
m
h
.
m
m
u
h
d
h
d
h d
Th e c o effi cien t o f th e firs t ter is n ity ; th e c o effi cien t o f th e s eco n
i s th e s u
th e c o effi c ien t o f th e t ir
ter
o f th e le tters a , b , c ,
e;
is th e s u
at a ti
te r
o f th e p ro
c ts o f t e s e letters t a en two
c t s ta e n t re e at
th e c o efii cien t o f th e fo rt te r i s th e s u
o f t e ir p ro
is th e p ro
e ; an d s o o n ; th e las t ter
c t o f all th e le tters
a ti
3
m
m
m
A ss
(
wh
m
m
.
um h h
e
a
x
t
at
)(
t
b)
x
.
du
m
u h
ese
.
(
m
wh d
la
ol
s
h)
x
m
"
x
1
ab 0
+
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h
du
in th e
+p l
c as e o
n" 2
’
az
m
k h
.
f
+ p 2 co
k
du
1 fac to rs
n
n
"
3
+p 3
x
n
t
4
‘
h
is ,
at
+p n
s
u
pp o s e
1
-
ere
=
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ab c
p n —l
u
h d
M ltiply b o t
s i es
by
ab d +
an o
h
t
h
er
o
( x + a) ( a + b )
“
w + (191 + 70)
n
x
ii -
1
—
su
mf
su
m f th
o
letters
su
m f th
pn
pro
a,
o
th e
_1 k =
e
n
n
n
in
o
n
p ro
b,
e
letters
du c t
—2
th e
all
k; t
s
( a + b ) (13 + k)
+ ( 192 + P1 k)
o
hu
fac to r a:
+
x
letters
du
c ts
k;
c,
p ro
a,
du
b,
f all th e
c ts
c,
n
a,
b,
k
ta
ii
—3
c,
two
en
k h
ta
en
t
. .
+
+ Pn —1 k
k;
at a
m
ti
e o
a,
b,
c,
k
f
all
m
ti
ree at a
ls ;
letters
‘
‘
.
e o
th e
f
a ll
XIII
C H A PTE R
B I N OM I A L TH E OR E M
161
(
+ a
33
my b h w
IT
.
e s
a
e
(
)
PO SI TI V E I N TE GR A L I
.
by
n
ac t
u l multipli
c a tio n
a
t
ND
h
EX
.
at
d)
+
x
.
1 cd
- -
x + a b cd
)
2
x
"
my h w v
.
w it d w th is u lt b y i p ti ; f t h
i ts f th
m f u mb f p ti l p
mpl p du t
u ltiplyi g t g t h f u
f w h i h i f md b y m
h
du t
f m h f th f u f t s I f w
b i g t k
l tt s
mi t h w y i wh i h t h v i u p ti l p du t
f md w t h t
i
m
h
b
h
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er
a,
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o
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er
t
a,
,
o
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a n
a,
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d
E
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l
m
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ar o
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o
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e
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n
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er a
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n
x
.
4
t
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ree
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n
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or
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e
o
+
en
en
(
2)
o
x
s
e
ro
e
x
(
x
+ 3)
(
x
n
e
e,
an
t
er a
s
e
o
e
o
.
er
a,
e t e r as
e
n
,
c,
c
o
e
a
—5
) (x + 9 )
(
+ 5x
t
one o
ou
o
e
e rs
.
-
3
e r as
e
e
a
a
e
)
4
x
ea c
o
.
— 135
-
a re
ac o r
n
a re
e
t
a c o r, a n
er
2
e
a
a: a re
n
o r
a re
a n
e re
o
n
ac o r
n
1
r
o
e
.
e
ro
.
ro
a
a
er
e
ac o r
r
ar
s
n
er
e re
o
ou
e
n
,
s
a
n
ac o r
er
e
or
a c o rs ,
o
on e
an
ar
n
c
s
n
c,
e
o u
s
er
an
4
a ou
c,
e
er o
n
a
.
t ree
an
ou
x
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e
O
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or
on
a
er
e
ec
n s
e
ro
n
a
e su
or
en
re s
n
o
s
a
e n
e se e
,
o
co n s s
c
e
n e
ex a
e
r
o
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,
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c
ro
eac
c s
e
e
o
—4 7 x 2 — 6 9 x 27 0
+
.
x
18
+ 27 0
—4 5
)
2
:e
B I N OM I A L TH E OR E M
d
m
(
mi v v
h
m
u
hu
u d
l
e
2
p
E xa
Fin
.
th e
x
POS ITI
.
fi c ien
c o ef
—3
I N TE
f x 3 in th e p ro
1 ) (x + 2 ) ( x
o
x + 5 (x
(
)
)
GR A L
du
I
ND
EX
1 37
.
ct
mu
md
3
in
t
VE
h
ltiply in g to ge t er th e x in an
Th e ter s n o l
g x are fo r e b y
y
e ric al q an tities o u t o f th e t wo re
th ree o f th e fac to rs , an d two o f th e n
fi c ie n t is e q al to th e s u
e n c e th e c o e f
ain in g fac to rs ;
o f th e p ro
c ts
1 , 2,
8 ta e n two at a ti e
3 , 5,
o f t h e q an titie s
T
th e
s
ire
re q
co ef
fi cien
(
a
1
( )
)
“
4
o
f t h e pre ce din g
4 ax
a
c
3
mplifi d
mt h d h
f m m g l u lt i
M t h mt i ; f it ft
h pp
Th e
e re
ex e
e n e ra
re s
o
o re
a
ro
a
cs
or
O
en e ra l
g
ro
s it io n
o
p
t
e
a
a
e
We
s
p
h
al
a
n o
or
x
or
o
e
o
as
n
h
en
an
a
:
2
4a
s
m
u
f u
o
h
v
mpl y t h
mi l Th
a
ra is e d
be
g
a rt ic le
a
we
s
u pp
o se
“
.
u
f de d c in g a p a rt ic la r c a s e
s o n e
of
re
n t
e
re n c e
i
n
c
o
c
r
q
a
e n s t a t it is
o re e a s
to p ro e
it is t o p ro e a p a rt ic la r c a s e o f it
t h e B in o
a c an
Ga
e
n e x t a rt ic le e
l in t h e
f mul k w
f th f m
w
p
a
— 16
.
u
x
t
— 40
I f in e q a t io n
w e Ob t a in
.
du
.
39
162
um u
u
m
k
m
to
o
eo re
an
m
y
v
v
u
mmh dt p
mb y wh i h y b i mi l
ig d p it iv i
l
g
.
e sa
et
e
n e
o
o
an
c
,
as s
y
u
ro
n o
os
e
a
te
n
e
ra
er
.
1 63
fi
To
.
n
d th e
in teger
C o n s ide r t h e
f
exp a n s i o n
wh en
o
n
is
a
o s iti ve
p
.
th e
n
u mb e
ex
n
f
a c t o rs ,
a n s io n
f
o
p
re ss io n
b e in g n
a c t o rs
as
is
p
l
n
i
t
o
e
t
p
g g
y
f
e x p an s io n
Th e
th e
r o
ex
f
n
.
h
u
u
f t is e x pre s s io n is t h e c o n tin e d p ro d c t o f
a, x
b , so c,
x
k, an d e e r t e r in t h e
lti
di en s io n s , b e in g a p ro d c t o r e d b y
le tt e r , o n e t a k e n ro
a c t o rs
ea c
o f t es e n
o
v y m
mu
u f m
f m h h
f
d i f md b y t ki g t h
m
h
s
f
i
Th h igh t p w
f m h f th f t
l tt
s i v lvi g
f md b y t ki g th l
Th t m
k
f m y l f th f t
f th l
d
b
i
f mth mi i g f t ; t h u h ffi i t f
th
t i
k; d
fi l p du t i th m f th l tt
b
by
Th
f md b y ki g h l t
m i v lv i g
k
f m y 2 f th f
f th l
b
d w
f mth tw mi i g f t ; hu h ffi i f f i
h
fi l p du t i t h m f th p du t f h l
er n
es
e
e r a:
e
ro
ro
an
e
ro
ro
n a
t
a
e
,
b,
n a
c,
n
ro
re
c
”“
a re
o
n
s
k t ak e n t wo
x
2
t
e
a re
e
at a
su
m;
ti
e
e
e
a
n
e
e t te r x
,
o
s
e
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t
e
ro
t
n
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co e
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.
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t
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t
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t
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e
e
n
c,
ta
o
c,
,
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c en
e
t
n
e t t e rs a ,
e
e rs a ,
t
o
e
a
co e
or
a c o rs
n
e
one o
a c t o rs , a n
e
a n
or
e
"'
or
.
s
o
s
an
,
a c o rs
ac o r
o
o
x
a c o rs , a n
e su
s
s
x
e
n
a n
n
e
n
O
c
an
ro
re
t er
e
ro
—
n
o
n
s
x
e n
o
ea c
er
e
er o
o
"
.
a
e
c,
"
g
n
e t t e rs
1 38
GH E R
HI
AL
G E BR A
.
f m
m
y
v v
f
m
f
a re
o r ed b
a kin
t
A n d, gen e rall , t h e t e r s in o l in g
y
g
—
n
n
r
r o f th e
a c t o rs , an d r o f t h e le t t e rs
tt
r
o
a
l
e
93
h
e
t e
y
t h e r re ain in g a c t o rs ; t
k ro
s t h e c o e fi c ie n t o f
a , b, c,
""
o f th e
ro d c t
h
l
rs
in t h e n al p ro d c t is t h e s u
o
f
t
e
e
tt
e
x
p
k t ak e n r a t a t i e ; den o t e it b y S
a , b , c,
f m
f m
fi
u
mi
Th e la s t t e r
H
"
S lx
I n S l th e
t
h
sa
at
m
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m
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is ,
02
in
u
S en
is
ct
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r
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s t it
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n
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Th e B in o
.
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h
v
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b u t in ( x a ) t e
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n is e e n o r O dd
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I f we
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on
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(
h
.
.
Th e c o e f c ie n t s in th e
165
ex p re s s e d b y
th e
e n ie n t l
co n
W e s all, o we e r, s o et i e s rt
n , a n d writ in g 0
02 , 0 3 ,
0
x
d
so o n
e
b e p ro e d a s o llo s
a c t o rs
B y in d c tio n we c a n fi n d t h e p ro d c t o f t h e n
a s e x lain e d in
w
a , a + b , 96 + c,
1
E
x
2
e
r
t
5
8
A
,
;
p
"
t e n de d c e t h e e x pan s io n o f ( oc + a ) a s in A rt 1 6 3
164
+
mi l Th
t
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at a
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m
m
s
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mb
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3
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es
T is is t h e B in o ia l Th eo re
is s aid t o b e t h e e x p an io n o f ( so
x
e
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_, x
t h in gs 2
n
"
s
n
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f
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er
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f
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t h e p ro d
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an
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yp
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v
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)
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a re
o s it i
v
e a c c o rdin
e
g
140
E
HI
xa
m
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f
de r
O f n e a ti
g
e
,
e
a
n o
w
c a se
a
o
c en
s
a
co n
u
re d
e
n e
our
c ed
a t t en
to
o n e
t io n
to
or
o
e
a
ac
n o
er
e
e
a s o
no
.
e,
on
mi l
.
r
n c e,
co
.
la s t c ap t e r w e in e s t iga t e d t h e B in o
o s iti e in t e e r ; w e s a ll
e n t h e in de x w a s a n
y p
g
lae t e re o b t a in e d o ld in t h e
et e r th e
o r
a n d ra c t io n a l
a l e s o f t h e in de x
IN
.
XI V
on
e
,
a
e
1
.
an
d by
a ct
u
al
v
di is io n ,
—x
l
(
)
an
d in
ea c
In t
h
h
o
ft
"
h
2
s
e se
we
e s e c a s es
e rie
hv
a
s th u mb
er o
e n
e
mp E 1 A t
f t mi u lim
it d
o
C
[
a re
er
s
x
s
n
b y in de pe n de n t p ro c e s s e s
h
r
,
.
e
o
.
.
b ta in e d
an
W
h
1
x
an d 1
e
f
f
t
e x p re s s io n s
x
o r eac
o
e
(
)
p
(
)
a rt ic la r c a s e s o f t h e
a re o n l
s all
re s e n t l
ro
t
at t e
e
p
p
p
i
of
1
h
re n
s an y
e n e ra l
l
a fo r t h e e x a n s io n
e
or
p
g
(
ra t io n a l
an t it
q
ex
h
an s io n
?
y v h hy
y
f mu
u y
u l w di v d b y N w
Th is f m
i
w h v tw
17 8
S u pp
p
w f uh
"
u
2
w
.
or
o se
.
p
o
e rs o
l
an
d
1
+
x, s
sco
as
a
a
e
+
n
‘
n L
.
re s s o n s a rra n
ed
g
as
c
m
m
(
m
x
o ex
e
to n
e
e re
(
n —
m( m
1)
l)
n
(
n
—
l)
1)
(
m
(
n —
2)
2)
x
+
in
asc e n
din g
1 50
HI
u
GH E R
G
A L E BR A
.
h
Th e p ro d c t o f t e s e t wo e x pre s sio n s will b e
e rs o f x ; de n o t e it b
c e n din g p o
y
w
h
h
s e rie s
a
in
m
fu
as
A , B , 0,
n c t io n s o f
a re
an d n ,
in a n y p a rtic la r
a n d t e re o re t h e a c t al va l e o f A , B , 0 ,
an d n in t a t c as e
ill de p e n d p o n t h e al es o f
c ase
Bu t
ic t h e c o e fi c ien t s o f t h e p o e rs o f x in ( 1 ) a n d ( 2 )
t h e way in
is q it e in dep e n den t o f
an d n
co
b in e t o gi e A , B , 0,
in o t e r wo rds , wh a tever va lu es In a n d 11
a y h a ve, A , B , O,
I
f
t
h
e re o re we c a n de t e r in e
reserve th e s a
e in va ria b le fo r
p
fo r a n y val e o f
an d n , we c o n c l de
th e o r
o f A , B , 0,
will h a e th e s a e o r fo r a ll va lu es o f
t at A , B , 0 ,
an d n
t
en
it is
c le a r
h f
w
h
at
u
u
wh h
v
m
us
v u
f
m
h
w
u
u
.
m
m
f
m
u m
mf m
v
m
f m
h
t
m
.
u
m
.
f
m
f
Th e p rin c iple h e re e x plain e d is o t e n re e rre d t o a s a n e x a ple
“
in t h e p res e n t c as e we
t h e p e r an en c e o f e q i ale n t o r s ;
of
a c t t a t i n an
l
i
r
a
e
b
a
a
l
ro du c t t h e
c
a e o n l t o re c o gn ise t h e
g
y
p
h
r
l
l
h
r
h
i
r
o
f
t
e
e
s
t
l
b
e
t
e
s
a
e
h
e
t
e
t
e
an t it ie s in
o
f
q
o l e d a re w o le n
b e rs , o r ra c tio n s
o s iti e , o r n e a ti e
p
g
m
uv
f m
hv y
f h
mw h
m
u w
v v
h um
f
v
h ll mk
W
f t h is p i ipl i th
h
Bi m
i l Th
mf
i
d
T
th
y
l
i
v
i
d
t
E
u
g
e
s
n o
e
s
e
a
a
o
r n c
o
e o re
a
ue
u se
e
or
an
ex
n
g
ro o
p
e
.
v
e n e ral
e
n
e
u
f
.
p
wh ic h
ro o
f
o
f
we
er
.
To p ro ve th e B in
17 9
vefrac tio n
i
t
i
s
o
p
.
mwh
mi l
o
Th eo re
a
the i n dex is
en
a
.
mp iti
mb l f (m) t d f th
—
—
m
1
2
1
m
m
m
m
(
(
)
(
)
)
m
+
Wh a tever
b e th e
n a l, le t t h e s y
r
a
t
i
o
c
f
l
t
h
en
f(
+
or
e s e rie s
or
s an
in tegra l
90
s tan
n
1
o r n ega tive,
ve
os
,
o
x +
l
w
i
l
)
n
f
va lu e o
+ nx +
d fo r t h e
(
— I
n
1
s e rie s
)
(
n
n
— l
1
2
)(
2
n
2)
-
3
i s t g th
t h p du t wi ll b
mu ltiply t h tw
wh
f
t
ill
s i i
i
w
b
di g p w
th
efi
m d my b
lt d i f mwh t
i b l f mf th p d t w my giv
i
t hi i
T d t m
h
l
t
t
m
t
i
t
f
t
i
u
s
v
u
h
t m d
;
y
p p
i
i
t
i
t
I
i
u pp t h t m d
v
t
h
m
p
g
f( )
i
m
f
1
d
h
f
f
i th
d
d
t
d
d
m
f
f
(
(
p
)
p
I f we
an o
er
u na
ere
s
s
1
(
an
o se
e ex
x
)
”
as c en
va
an
e
dt
or
h f
e rs
o
e
n
o
e re o re
o
o
e
or
o
co n
a re
os
a re
os
e
an
a
n
ro
e
o se
x,
11
a
er
e
an
n va r a
s
an
a
n
a ever
n e
an
n
an
n
or
n
e er
o
o
er es
s er e
o
e se
e
c
c en s
co
e
e
.
e
e
n
ro
en
en
e rs
s
uc
.
e
or
n
e ex
a
s
s
an
e
r
o se
c ase
e
or
o
1 52
HI
GH E R
To p ro ve th e B in
180
n ega ti ve qu a n ti ty
o
.
AL
G E BR A
mi l Th
a
.
mwh
eo re
en
i n
dex is
an
y
n
is
.
I t h a s b e e n p ro
v
ed
t
h
at
m
m
f( ) f( ) f(
)
R pl i g mb y
f m d
x
fo r
v lu s
a ll
s it i
o
p
v
e
)
,
o
e
a
we h a
v
an
n
n
n
e
.
ac n
n
wh
(
e re
e
f(
-
n
) f( )
n
x
—n +
f(
:
=
n
)
f ( 0)
1,
s in c e a ll
m
ter
s o
f th e
s e rie s e x c e p t
1
=
f (n )
but
f( )
an
n
yp
f(
o sitive
1
1
(
+ x
)
1
(
Bu t
f
n
)
s t an
wh i h
c
H
en c e
ro
v
p
the t
h
)
eo re
(
mi
o
s co
f
rs
v lu
f(
a
e
_ n
)
”
is
h
)
_ n
f(
an
,
l
s e rie s
x
t h e B in
es
=
ds fo r t h e
—n
"
fi tv
th e
“
mi l Th
mpl t ly
a
e e
n
— 1
e o re
)x
2
mf
e s t ab
+
or
lis
h
an
y
v
n e ga t i
ed
in de x
e
.
.
m
m
Th e p ro o c o n t a in e d in t h e t w o p re c e din g a rtic le s
a
y
n ot a
ar
ll
ro b a b l
re se n t s o
i
f
e
o
s
a t is a c t o r , a n d will
e
d
pp
y
p
p
fi c u lt ies t o t h e s t de n t
T e re is o n l o n e p o in t t o
ic
we
S a ll n o w re e r
181
.
wh y
u
f
h
f
.
h
y
y
wh h
.
m
f
um
u m
y
u
mi fi
I n t h e e x p re ss io n fo r ( ) t h e n
b er o f t e r
is a p o s it i e in t e ge r, a n d n li it e d in a ll o t
A rt 1 8 2
I t is t e re o re n e c e s s a r t o e n q ire in
m
v
.
.
h f
s
h
s
n
ite
en
See
a t s e n s e we
er c a se s
wh
wh
.
B IN O M I A L
TH E OR E M
m
ANY I
.
ND
EX
1 53
.
h t f (m) f ( ) f ( m ) I w ill
b
i Ch p
t h t wh
h f t h i f (m)
1
i
m
d
m
i
h
t
u
i
h
m
i
l
(
f
g
f(
f ) (
)
)
i
l
u
v
t
f
m
B
t wh
ll t h
i
1
f ( ) f( )
q
w
d
l
i
w
y
h
u
y
di
t
t
f
l
i
l
h
m
t
i
g
p
d
t d b y f (m
b
t
h
i
s
b
d
t
d
h
t
fi
t
m
t
) y
y f( )
f t h p du t will g
wit h t h
fi t t m ff
wh t v
fi i v lu my h v [S A 308 ]
re a rd
g
to
a re
e s een
n
a
n
n
,
ver en
en o
s ta te
t,
a
s c o n ver en
t,
e
an
c an
ro
a e
er
xa
c
a s se r
te
n
m
pl
1
e
a
e r
E x p an
.
(1
x
d(
xa
e
a
a
e
3
1
x
)
?
to fo
u
r
m
l
p
2
e
2
.
x
+
--
8
x
2
E xp a n
1
+
E
3
x
o r
se r es
rs
s
er
a re
ser es
r
er
s
o
.
.
t )(
3
1
e a
3
79
—2
.
)
2+
”
+
d
to fo
-
u
5)
(2
3x
m
ter
r
s
.
5)
3
1+
—4
2
-
6)
3
2
)
n o
w
3x
1
45
1
1 82
la
f mu
s
3
2
3
,
et ca
e
e
,
m
ter
3
3
)
es e
t
r
rt
ee
.
(
(
rs
t
e ar
a
n
-
1—
E
re e
tr
e
e
t
.
e s er es
e
,
a
en o
o
s
x >
en
t
n
eac
,
n
on
a
n
<
x
an
u
3
"
en
e se r e
e
x
a
n
x
e
o
t t
en
t e r XXL ,
o
a en
e
E
th e
In
.
fi
n
n
di ng t h e
(
n
—l
)(
n
e n e ra l
g
-
mw mu t
ter
2)
(
n
e
s
th e
u se
.
—r + 1
)
E
w itt
wh
en
r
en
n
in
is
f
ym
v
fu ll ;
fo r t h e s
bol
ra c t io n a l o r n e a t i e
g
O
can
n o
r
lo nge r b e
e
mpl y
o
e
d
.
m
v v hu
w h
h
wh
u y
v
hu
m wh
n le s s
A ls o t h e c o e ffi c ie n t o f t h e ge n e ra l t e r c a n n e e r a n is
ill t e re
a c t o rs o f it s n
e ra t o r is z e ro ; t h e s e rie s
o n e o f th e
‘h
en
t er ,
en
o re s t o p a t t h e r
n — r + 1 is z e ro ; t a t is ,
1 b u t s in c e r is a p o s it i e in t e ge r t is e q a lit c a n n e e r
r= n
s th e
T
o s it i e a n d in t e gra l
e n t h e in de x n is
o ld e x c e p t
p
en
1 ter s
ia l T e o re
e x t e n ds t o n
e x p a n s io n b y t h e B in o
s in a ll
e r o f te r
b
o s it i e in t e e r, a n d t o
an in
n it e n
n is a
g
p
f
f
h
o
t
h
wh
v
e r c as es
.
um
m wh
m
h
v
m
fi
v
h
.
um
m
HI GHER
1 54
m
E x a p le 1
Fin
.
d
(r +
Th e
n
v h
w
ti
e
th e
th e
n
Th e
xa
m
mb
u
ab o
um
f fac to rs in th e n
1 out o f
b y ta in g
ex p res s io n
er o
(
m
pl
e
1)
r
‘h
f ( 1 + x)
1
v
e
k
is
of t
erato r
eac
h
2
Fin
.
d
r, an
h
ese
d
1
n egati
r
h
f t e s e are n ega
e fac to rs , we
ay
v
o
mi
th e gen eral ter
th e
n
e x p an s io n o
m
f (1
7t x
)
— 2n
1
(
)
(1
'
-
;
1
ter
n
my
1
— r
+
-
r—
l
n
.
)
"
)
If
.
(
1M
(
n
n
l ) ( 2n
-
-
r—
1)
(
l
.
n
—1
)
[
r
—l
)
(2 n
-
(
1)
r—
L
—l
)
n
l
l
—1
)
m
E x a p le 3
Th e
(r + 1 )
th
Fin
.
d
mi
th e gen eral te r
m
mvi
o
n
1
e x p an s io n
-
r+
o
f
1)
r
4
5
(
r+
1
re
-
—3
)
3
by
2r
th e
n
—5
ter
k
g li
e
fac to rs fro
mth
-
2)
r
e n
m
—3
2
r
(
)
5
E
ex p an s io n o
1
1
.
ter
; t ere fo re ,
rite
GE BR A
mi
th e ge n eral ter
1
Th e
AL
um
erato r an
d
d
en o
mi
n ato r
.
1 56
H I
u
wh
u qu iv
d, if n o t, n de r
b e u s e d a s it s t r e
an
at
e
u
S p po s e , fo r in s t a n
1
(
h
in t is
ta
k
e
u
q
h
T is
i
t
p
at o n
u
— x
l
a en
t
)
h
re s
GE BR A
ditio n s t h e
t
ex
.
p
an s io n
f (1
o
)
a
s
"
my
a
.
1 5t
at n
1
‘
AL
h
hv
we
en
a
e
1
2 5 we t
a:
y
h
o
en
b t a in
u lt suffi i
c en
1s
t to
s
hw h
t
e
we
at
t
c an n o
e
1
as
co n
c e,
t ra dic t o r
co n
GH E R
(
n
+ nx +
1
2
h mti l quiv l f ( 1 ) i
f mth f mu l f t h m f
w
k
w t h t th m f th fi
t h e t ru
e a rit
N ow
re s s io n ,
g
e
ro
e
o
or
a
o r
n o
e
t
a en
e
ca
su
l
— x
l —
“
x
e
e
a
l)
s e rie s
1)
n
su
o
o
a ll c a s e s
n
e r c al
g
rs t
e
mt i
t m
eo
a
.
er
x
s
o
p
ro
f th e
”
x
1
l
an
wh
d,
en
la rge we
s
a
u ffi
t
as
we
re a t e r
g
n o
an
c an
c ie n
litt le
t
h
n
an
u h pp
m
b
u
y
s
c
n
is
a;
a
n
u m i lly l
l
e
a
s
e
p
1 , th e
as s
er o
er
k
1 , b y t a in g
an
as
a
s
er
e
su
e
e as e
9
s
u ffi i
s,
a
e
c an
u
ro
c en
y
tl
a
en
e
s
a:
o
n
a
er
n
as
er ca
a
c
v lu
a
s o
e
o
a:
f
l —x
to th e
on
f th e
1
w ill
h
°
x
mti
ft m
a
t
x
-
w pl
ll
m
t h t i b y t ki g
1
f t m th
m b md t diff
1
f ml —
B t wh
i
u m i lly
x
e
1
r
er o
ro x i
ess
er ca
mk
u mb
a
— x
3
v lu
a
in
e o
c re a e s
s
wit
h
f
is
b t a in e d b y ta kin g
o
r,
an
dt
h f
e re o re
s e rie s
x
x
2
m
a
h
v
w ys
y
v
m h m
hm y
an d D i er
b e s e e n in t h e c ap t e r o n C o n e rge n c
t
a t t h e e x an s io n
b
t
h
i
a l T e o re
e B in o
e n c y o f Se rie
p
y
g
i
n
a s c e n din
rs o f a: is a l a
a rit
t
i
a ll
i
n
o f
x
o
e
e
c
1
+
g p
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hm
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l
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f
ra c t io n
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m
m
m
mi
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m
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ha ra cteris tic of th e lo ga rith
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G A R ITH M S
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mm
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m
h
f
h
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w
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e
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t
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)
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e
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i
t
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n
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h
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n
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b er of c ip h ers i
edia tely after th e dec i
a l p o i n t, a n d is
n ega tive
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m
m
mm
m
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s hy
t o b a se 1 0 o f a ll in t ege r
Th e lo ga rit
ro
212
1 to
2 00000 h a e b ee n o n d a n d t ab la t e d in
o s t Ta b le
t e a re
als
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ults
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b e writt e n do n b y in s p e c tio n , s o t a t o n l
t o b e re gi t e re d in t h e Tab le s
re s
alre a
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Le t
b e an y n
213
t h e p o s it io n o f t h e
e re l
a lt e r
di idin g b y a po e r o f 1 0
de c i a l p o in t
it o t c h a n gin g t h e e q e n c e o f g re s , it o llo ws
”
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n
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.
v
m
h
x
m y
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In
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m
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s
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m
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H I GH E R
1 82
AL
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L et
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I n th e Tab les
e fi n d t at 38 59 6 3 6 is th e
deCI al p o in t as ell as th e c arac teris tic b e in g o
th e c arac teris tic o f th e lo garit
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a ra c ter s t c , an
e
e
n
e
c as e
e c
,
re c e
a
ro
a ra c e r s
an
so
e
n
.
a rran
r a e s
ro
a
.
r
,
os
ts
e
e
s sa
t
ree
e
os
o se
s
r
t
an
f lo g 1 6 5
HI G HE R
1 84
h
T is
re s
ult my
a
a ls o
b e p ro
x
lo g b ,
L et
t
h
en
b y t a k in g lo ga rit
v
G E BR A
ed
dire c tl
as
h
b
at a
t o b a s e b , we
”
hv
a
o
o
s
e
lo gb b
lo gb a
x
.
y f ll w
t
so
a
h ms
AL
10 n
lo gb a
x
l
.
m w
u
hm
u
hm
m y
f w
f
y
u
ill ill s t ra t e t h e
2 18
t ilit
of
Th e o llo in g e x a ples
a rit
a c ilit a t in
e t ic a l c alc la tio n ; b u t fo r in
lo ga rit
in
g
as
ic Tab le s t h e re a de r is
t o t h e u s e o f L o ga rit
o r a t io n
et r
re e rre d t o
o rk s o n Trigo n o
h ms
.
f m
f
w
m
4
v
u dv u
E xa p le 1
Th e
.
req
Gi
.
ire
en
fi n d lo g
lo g
al e
= 3 10 g
27
4
I0
g
lo g
81
5
16
10
1
4
lo g 9 0
4 —
—
l
o
3
2
( g
)
3
l
3 ( og3
97
—
x
5
1 00
lo g 3
5
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l
o g3 4 1)
(
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5
8
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2 7 7 8 07 6 6
stu d
Th e
po
s
we r
sh uld
bt
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en
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a
t
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a
lo g 5=
m
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Fin
.
n o
o
d
lo g
th e
lo g 2
lo
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g (8 7 5
)
1Q
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n
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a n e
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h
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ro
um
d
8 01 0300, lo g 7
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rt
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16
th e
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— 1 6 lo
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s
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at
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m
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en
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hm
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x
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8
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o
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th e
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hv
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4 x ) ( lo g 2 + lo g 3 )
-
185
.
d lo g 3 , fi n d to t wo p lac e s
h d
fb o t
3
(
(3
G A R ITH M S
8;
(
+ 5) 2 lo g
x
— 4 lo 2
g
lo g 2 ;
— 3 lo 2
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3 lo g 3
l 0 10 g 2 ;
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4 44 16 639
2 51 054 52
EXA M PLES
1
Fin d, b y in s p e ctio n , th e
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2 1 7 3 5, 2 3 8
c
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-
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.
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.
.
arac teris tic s
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dw
Th e an t is s a o flo g7 6 2 3 is 8 8 2 1 2 59 ; write
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,
,
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,
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t h e lo garit
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o
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e
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en
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o
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lo g 2
in th e in tegral p art
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2 7 7 8 1 51 3 ,
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v
h
1 4 7 7 1 2 1 3,
4 3 01 03 ,
4
t
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fi
3 6 9 89 7 ,
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0 9 1 08 1 5,
lo g 3
fi
i
f th e
n
umb
in t h e
n
u mb
o
e rs
5 6 51 5 ?
c an t
fig
u re
54 8 7 13 8 4
4 7 7 1 2 1 3 , lo g 7
e rs
.
8 4509 8 0, fi n d t h e
f
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n
12
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d th
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t
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v
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en
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13
.
lo gi/0 1 05
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h vi g giv th
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en
n
7 6 44 3 6 3 6
2 2 8 9 6 8 8 3 , fi n d th e
at
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e le
v h
en
t
ro o
t
o
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HIGHER
86
16
v h
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g
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n
.
t
en
at
d
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g
v
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ct o
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an
18
.
19
.
C alc lat e t o
lo g 2
en
an
u
a
n
d 10 g 1 9 1 56 3 1 = 6L2 8 2 3 l 2 0
.
d lo g 3 , fi n d lo g ( 3 48
s ix
u
C alc lat e t o
.
an
h vi g
d lo g 3 , fi n d lo g
lo g 2 , lo g 3 , lo g 7 ;
20
.
1
Gi
en
GEBR A
f 3 7 2 03 , 3 7 2 03 , 0 03 7 203 , 3 7 2 030,
1 5 7 057 8 0,
en
AL
d iml pl
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als o
s ix
a
x
v lu
th e
a c es
3
1 08
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e o
f
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p
o
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ec
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e o
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f
22 x 7 0 ;
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i
g
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en
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'
21
.
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d
th e
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n
umb
er o
l
g
f
.
S
.
D e ter in e h o w
22
23
Fin
t
v
So l
27
29
d th e
fi
d
f igits in 3
my
an
ifi c an t
wi g q u
n
e
d
is greate r t
)
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th e fo llo
e
3
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at
m
an
24
t
e
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h
p
ci
h
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atio n s ,
28
.
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.
h
e re
1 00 0
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a
n
26
28
=
~
.
30
6y
.
2
x
.
.
32
.
v
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li
ml
th e de c i
a
6
x
—2
=
an
5—3x _
5
.
2x
5
.
7
d lo g 7
2
x +
.
2
1 —x
—
—
x
1
v= 4
y
3
_
3y
296 — 1
~
3
2
31
ee n
lo g 2 , lo g 3 ,
en
.
x + y
w
t in
25
x
2
x
an
t
e rs
.
.
12
en
10 g10 2 = 8 01 03 , fi n d 10 g252 00
en
10 g1 0 2
.
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=
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8 4 509 , fi n d 10 g7 J 2
an d
10 g,/2 7
.
HIG HE R
1 88
h
th e
en c e
1
i
s th e
( )
se rie s
93
"
1
p
AL
GE BR A
.
o w e r O f t h e s e rie s
2
( )
t
h
at
is ,
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[
I
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in de n it e l in c re a s e d w e a e
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fi
y
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l
cc
+ x + 7
+
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e
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f
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s
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096
fo r
x,
l
t
h
x
+ x +
3
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+
B
+
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9
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=
1
so
+ x
0 90
|
t
h
at
c
z
3
lo g a ;
e
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s t it
uti
n
f
o
g
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m
h
.
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n
is in
fi
n
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mit
it e , t h e li
o
f 1
1
+
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ee A r t 2 6 6
[
.
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r c
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T is is t h e E xp o n en tial Th eo re
C OR
4
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l
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|
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1
4
33
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n
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13
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é
h f
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L
ted b y
If t
.
1
— 4
3
en o
a
e
1
s e rie s
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b
a
M E L
Th e
is
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my
n
s
in t h e pre c e din g in v e t igat io n , it
is in de n it e l in c re as e d,
as
fi
y
x
2
3
1
x
3
6
x
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4
1
my b
a
e sh e
]
wn t h at
E X PON E N TI A L A N D
t
h
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is ,
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en
is in
n
fi
u
G A R ITH M IC
mit
ite , t h e li
n
1
By p tt in g
LO
hv
we
7
a
7 1.
f
O
S E R IE S
1 89
.
1
e
—
m
r
.
Now
th
us th
H
a
v
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s ed
s v
e r es
en
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v
i
g
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t ice
f
S in c e
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fu
is pla c e d
up
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ca
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ro o
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n
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r
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fi
n
v
a
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o
f
v
th
it e , we h a
x
r
mit
t h at t h e li
is
re s t ric t io n
n o
le s s t
Is
e
a
u
o
l
er
,
t h at
a rt ic e
ar
u
I S c le a r
ite ;
n
e
de n o te t h e
us
n
h u ity th p i w
it h mt i lly i t lligib l
A
t
1
8
3
[
]
i
i
i
h
i
h
th
t
t
f
f
wh
p
g
g p
i i fi it
ssu md t h t w h
W h v
an o
.
fi
z
f
o
s
u lts
is
e re
f
It
is in
n
w
t h e li
o r a ll va lu es o
Le t
wh
it e
n
I n t h e p rec e din g
1
in c e
o f 23 ; a ls o
Bu t t
de
n
o
.
v lu
th e
s
en ce
221
ha
e
mi i fi
it f
lim
m
an
e
o
at
f
u
is ,
2
y
is
u
,
mit
;a
d ge n e ra ll t h a t
h en c e t h e li
Of u
is
O f 2 03
is
90
H I
22 2
Th e
.
GH E R
AL
GE B R A
.
s e rie s
1
1
—
—
L L
li
1
wh i h w h v d t d b y i v y imp t t it i th b
t
wh i h l g i h m fi t l u l d L g i h m t t h i
b
k
t h N pi i
y t m md ft N pi
w
l
ll d
th i i v
l l g i hm
s f mth
t
Th y
f t t h t t h y t h fi t l g ith m whi h tu lly m i t
id
lg b
i l i v t ig t i
i
ti
s
u d i th ti l w k it i t b
Wh
l g ith m
i
lw y u d
mmb d t h t t h b
t
d ju t
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l
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it h mti l w k t h b
1 0 i i v i b ly m
p
F mt h s i t h pp imt v lu f
b d t m
i d
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l
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t
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f
d
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m
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d
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y
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m
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o
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n
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ase
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en
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o
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ase
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re e O
e
E x a p le 1
.
in
th e
O f th e
su
1
fi n ite
—
l_
h ave
l
.
an d
u
b y p ttin g x
1 in th e
e
h
en ce
E
th e
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e
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m
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m
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th e
Fin
d
1
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fo r
+
x
e
,
= 1 —1 +
s e ries
th e
ax
s e rie s
1
is
Q
"
c o effic ie n t o f x in th e
x
2
)
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e
— ax — x
1
(
e xp an s io n O f
x
2
3
1 ) rx r
,
er
a
e
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+
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o
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as
n
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ec
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I
s
s
e
e c an
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co
or
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1
o
ro
ra
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n
o
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er
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We
ar
s
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e
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e O
a
t
e
.
ar a
n
ra c
acc
s
a
a e
ro x
e a
n a
o
s
ar
n a
e o re
a
s
ase
on s
a
s
so
c
n
e
,
s
es
o
n a tu ra
e
as
.
s e
ar
o
n
e
re
re
ca
ra c a
e
a
ca
e
e
a
ar
e re
ar
o
n
s
er an
rs
e
at e
ca c
a re a so
e
o r an
er
rs
a
e
a re
e
on
e
as
s
e,
a re
s
.
e ra
e
t
o r
a
co n s
en o
e er
a s
n e
s
192
AL
H IGH E R
G EB R A
.
wh
v y m
um
u
h
a ll t h e s e rie s fo r lo
er
s
en a
o is
1
22 4
E x c ept
x
+
e
g(
)
W e c an , o e e r,
e ric a l c a lc la t io n s
is o f lit t le u s e fo r n
it o t e r s e rie s b y t h e aid o f w ic h Ta b les o f L o ga r
de d c e ro
a
b
e c o n s t r c t ed
it h s
y
.
f m
u
mm
u
B y wrl tl n g
l
fo r
ii
B y wrl t
si
mg
bot
n s on
g
h
mw
e o
1)
fo r
ro
lo ge ( n
e
1
it
2a
b t ai n
u
q
a t io n
1
1
n
n ce, b
h
e
5
y
c
h
an
in
g g
,
l)
E
ditio n ,
ad
1
1
1
77
571
.
,
f mu l
ce
1
1 ) — lo ge ( ri
F mt h is
h
n
h
e
5
I
o
lo ge ( ri
d ( 2) b y
an
+
we
f th e
lo ge n
F m( 1 )
a
c
b t a i n lo g,
lo ge ri
7 7.
s ide s o
hwv
.
lo ge ( ri
1
.
u
b y p t t in g n = 3 we o b t a in lo ge 4 — lo ge 2 ,
f
t h a t is lo ge 2
an d b
i
n
h
a lc la t io n we fi n d t a t t h e
e
e
c
t
t
e
c
y
g
wh e n c e lo ge 8 is k n o n
v a l e o f lo g6 2
6 9 3 1 47 18
or
ro
u
a
f
u
w
°
u
A ga in b y p t t in g
fi n d lo g3 1 0
9 we
rt
o
.
wh
lo ge 8
b ta in lo g6 1 0
h
en c e
we
v t N pi i l g ith ms i t l g ith m t b s 1 0
1
w m
u ltiply b y
h i h i th md l
w
A
t
2
1
6
f
h
t
[
]
l g 10
1
mm y t m d v lu
4 3 4 2 9 4 48
2 3 02 58 509
w
h ll d t th is mdulu s b y p
To
co n
er
e r an
a
e
o
co
on
e s
s
s e
en o
a
,
o
s
c
’
o
n
ar
e
u
o
ar
o
o
s
a e
r
u s
o
e
6
an
i ts
e
a
e 1s
o
o r
.
I n t h e Pro ceedin gs of th e R o y a l S o ciety of Lo n do n , V o l XXV I I
a e 8 8, Pro e s s o r J
l
A
a
h
a
i
n
t
h
a
0
d
s
s
e
e
e
o f e, ,u ,
p g
g
lo g,3 2 , lo g, 3 , lo g, 5 t o
o re t an 260 pla c e o f dec i als
f
.
mu
m
.
m
h
y
v
v s
v us
m
s
h
u
u
.
.
.
I f we
lt ipl t h e a b o e e rie s t ro gh o t b y p , we
2 25
lae a dapt e d t o t h e c alc lat io n o f co
or
o n lo ga ri th
o b t a in
s
M
L
ro
Th
l
l
o
o
1
n
n
a g, (
3
) p ge
372
n
2 7 1:
f mu
.
us
f m
u
.
,
mm
m
.
E X PO
t
h
at
N N
E
LO
AND
T IA L
SE R IE S
19 3
.
is ,
lo
(
g
D
M
10 8
-
M
“
lo g
F m it h
e
ro
f one
th e o t
.
2 73
M
u
1
+
3 71
3
ro
ar
o
o
4
e
“
n
mil ly f m
Si
G A R ITH M IC
er o
f t wo
— lo
go
ri
(
,
—1
n
7
ab o
t
c o n sec
e
+
2n
70
ul
u iv u mb
f u d d th u
f th e
v
)
t s we
e re s
n
e rs
e
,
2
a
3n
see
b e kn o
a t a b le
h
w
t
at
n
+
hm
if t h e lo ga rit
t h e lo ga rit
hm f
o
h my b
f l g it h m
b
t u t d
I t h u ld b
m k d t h t h b v f mu l
ly
d d
l u l t t h l g it h m f p im u m
b
h l g i hm
f
h
f
m
i
m
m
i
u
b
b
t
i
b
d
d
t
t
h
b
d
p
g
y
y
g
l g it h m
s f i mp t f t
I
l u l t t h l g it h m f
d t
h
m
ll
f
t
y
w
b
l
l
i
m
i
u
m
d
t
u
u
y
u
b
t
i
u
h
u
m
i
h
t
t
t
b
p
f th f m
ul ( 1)
b t w
m v lu
d v u t fi d
b y wh i h div i i
my b
ily p f md
f
d u h h t
—l
it h
1
t i s t h giv
u mb
f t W
h fi d l g ( 1 ) l g ( 1 ) d d du t h l g it h m f
h giv
u mb
u
v
m
u
v u
hu
w h v
er
co n s r
c e
s
to
ca c
o
a
o
o
e re
ar
a e
e
ar
te
os
o
n
t
e
en
n +
o
s
ar
c an
e
e
or
o
n
n ee
e
o
o
e
e
t
ar
t
er
e
e
eas
en
o
er
e n
o
n
er o r
e
an
s
er a s a
n
an
ac
e
ce
e
er
n e
t
er
e
a
e
c
t
s
so
r
,
a
e
on e o
an
ea
en
n
t
o r
a
s
e
a n
e rs ,
o
s
e
ae a re o n
or
.
ar
u
a
e
a n e
a
s
Calc late lo g 2
B y p ttin g n
:
1 0 in
an
e
d lo g 3 , gi
a e
0 4 3 4 29 44 8
2 lo g 3
o
a
or
e
.
o
ar
th e
en
al e o
n
4 lo g 3
3 lo g 2
:
th e
we
8 0 in
ob
0 0000007 2
0 00000006
tain lo g 8 1
3 01 029 9 9 7
.
hu
lo g 8 0 ; t
0 00033 9 29
lo g 2
s
0 000002 8 3
0 00000003 ;
0 053 9 503 2 ,
.
h
v
h
f
o 1
s e 1 ie s
an o t e r
i
e
g
se
l in t h e c o n s t r c t io n o f
Fo r rt e r in o r a t io n o n t h e s b e c t t h e
’
M r G la is e r s a rt ic le o n Lo ga ri th s in t h e
we s
i l
ic
is o t e n
a rt c e
wh h
.
s
0 000008 6 8
0 08 4 8 5024
lo g (n + l ) — lo g n
L o ga rit
ic Ta b les
re a de r 1 8 re e rre d t o
E n cy c lo p cedia B ri ta n n ic a
hm
f
lo g 9 ; t
f lo g 1 0
0 0001 08 57
3 lo g 2
n ex t
.
0 001 4 4 7 6 5
0 054 2 8 6 8 1
1
~
0 02 1 7 1 4 7 2
lo g 3 : 4 7 7 1 2 1 2 56
P ttin g
=
4 3 4 29 448
n
2 lo g 3 : 0 4 57 57 4 8 8 ,
1
In
o
.
.
u
o
er
n
E x a p le
1
o
o
o
e n
a c o rs
n o
co n
or n
en
r
e
e
s on
c
e a
o
a
o r
a
e
er n
s
a e
e
t
a
o n en
ca c
o
or
t
s
e
co
e rs ,
n
e
an
,
er
ts
o r er
e
o
n
o
e n
o
n
o
co
n
o
.
ar
r
e
a
,
.
f
fu h
h
a ll
u fu
f m
u
uj
m
1 94
H I
226
GH E R
AL
hv
ro
I n A rt 2 2 3 w e
e
a
.
.
p
G E BR A
v
ed
g
g
x — —
h
c
an
i
n
g g
a:
in t o
hv
we
a,
a
—w
)
By
s
ub
t ra c t io n
t
h
.
at
3
$
3
e
=
,
2
c
lo ge ( n
+
N OTE
T is
co n
so
h
1 ) — 10 ge n
v
e n ie n
227
a
.
h
t
Th e
.
er
t
p
h
th e
s erie s
o
o
m
n
hu
b t a in
s o
1
1
J
5( 2 n
v
erge s
in A rt 2 24
.
g
ex a
d
e ry rap i
1
- -
r
1)
ly , b u t in p rac tic e is
n o
t
w
al
ay s
.
mpl
ut
ill
es
If
.
B
a,
are
th e
ro o
ts
o
f th e
a
10 g ( a
at
Sin c e
th e
ra t e
s
a
°
+ ,8
1
—bx
b x + cx
_ lo g a +
m
pl
e
2
we
9
lo g ( a
xa
2
eq
u
s
ub j
ect
atio n
2
+B
x
2
a
E
1
v
f ll w i
we t
as
o
f th e
.
E x a p le 1
t
as
at
2
:
s eries c o n
h
t
so
.
Pro
lo g ( 1 + x + x 2 ) is —
?
lo g ( 1 + x + x 2 ) = lo g
+
ax
v h
t
e
3
—
1 x
th e
d
a
1
{ +
a
( l + ax)
3
+
x
3
i
3
as n
)
x3
is
t
o
or
is
lo g ( 1
2
%
—
c o ef
fi c ien
a
ax
+fl
3
x
3
fl }
x
2
lo g ( 1 + 7323)
)
2
c
3 3
x
—+ fi
f
"
x
in
n o
t
x
a
th e
multipl
e x p an s io n
e o
f3
.
)
x
x
'
3
9
x
3
+
3
a e
in g
lo g ( 1
— x
6
at
1 acc o r
71
L
x
)
3
hv
= lo g a + lo g ( 1 +
9
or
l
2
2
a
s
+
2
x
3
"
x
hw
e
196
HI
3
S
.
hwh
t
e
GH ER
s
h w th
e
x
.
hw h
t
e
—
o
f
d iml
7
a s
Pro
.
lo
9
b
1
—
a
Q
B
b
lo g, a
a
th e N ap ie rian lo ga rit
v h
v h
e
t
at
hm
e
t
at
—x
d
v lu
th e
a
2
e
f
.
d
—
)
=
11
v
Sh w th
e
.
a
1
2
g +
(
| Z
—
um
4
x
—
y
4
°
at
if ax
?
Pro
2
v h
e
an
t
d
a
+
o
5
2
Pro
v h
e
t
1
E x p an d
lo garit
on
hl u
es s
n
lo
5
o
hm f 7
s o
ity
2
2
1 + ax + a +
g
.
5
2
9
6
4
3
g 5
.
+
e s e rie s
.
4
5
1
3
6
2x
m f th
t
.
mm
co
l + 3x
d fin d t h e ge n e ral ter
14
l
ac es
p
G
at
10 g‘
an
S ix t ee n
3
m f th
.
to
at
d fi n d th e gen e ral t e r
13
(x
f th e
are e ac
—
an
1
)+
v u
2
.
c o rre c t
f
e o
—
12
.
2
erical
al es
Fin th e n
10
an d 1 3 ; gi e n p = 4 3 4 2 9 44 8 , lo g
.
lo g o
e
l
e
g(
Fin
.
x
4
.
Pro
.
8
d
n
ec
2
3
at
a
Fi
.
2
I2
a
.
)
=
2
at
S
5
6
a
If
.
GE B R A
at
lo g, ( n
4
AL
a
6
33;
o
m
2
e s e rie s
a s e ri e s o
4
3
.
mg p
f as c e n d
o
w
e rs o
fx
.
,
E X PO N E N TI A L A N D
15
E x pres s
16
S
.
.
hw h
t
e
l0 g ( U + 2k)
2 10 b
e
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e
/l )
(
3;
ro o
If x < 1 , fi n d th e
a
2
.
2
S
.
2
+ 10
ts
o
h w th
e
+
x
o
2
2
3
3
2
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m f th
su
,
w
p
o
th e
e rs o
f
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l
x
e
4
e
.
-
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hw h
f m m+
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21
or
2,
an
t
at
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Pro
.
v h
t
e
at
d
2
3
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4
be
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s
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d
c alc
ul
at e
p
an
fi
o
.
1
.
hw h
t
e
4
-
3
at
3
x
3
5
3,
5
23
t
1
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e
o
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s e ries
a
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is
o
3
3
4
.
3
10
(n
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1
152
1
1
1
2n
24
a,
1
+
3
1
1
9
1
lo g. —
25
lo ge 3 = 1 1 a
2
I) , lo g,
81
80
c, s
hw h
t
e
at
3 b + 5c, lo g, 5= 1 6 a
lo g. 2 , lo g. 3 , lo g, 5 t o 8 p lac es
o
ml
f dec i
a s
.
f
be
If n
m4 m
f th e fo r
— lo
ge
lo g 2 = 7 a — 2 b
,
3
.
c o e f c ien
n
.
(1
3
;
3
4
.
2
l
24
J
/l ) 6
3(7
at
Sh e w t h at
.
i=
e s e rie s
3
+
Z
S
2
+ 1)
if n b e
2 lo ge n
23
(
n
th e
t
1+
22
fl )
4
l
1
I f lo g
e re
at
2
.
wh
1 97
.
4
2 (x
f x —p x + g= 0,
1
20
fx,
e rs o
2
B)
+
1
19
o
S E R IE S
.
.
.
n
0 b0 e ( I)
10 s ( 1
18
di g p w
as c e n
[1
1
_
d B b e th e
an
G A R ITH M I C
at
0
’
17
in
LO
as c en
o
dd,
di g
n
o r o
f
X VI I I
C H A PTE R
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ANN
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.
h
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229
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n t e re s t a n d D is c o
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q
lae
b y t h e u s e o f alge b ra ic a l o r
.
u
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h
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f mu
mI
u
h o w t h e s o l tio n
n t
a
li
si
b
e
y
p
u m
m
fi
f
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o
.
D is co u n t, Pres en t Va lu e in
e t ic a l s e n s e
b u t in s t e a d o f t ak in g a s t h e
t h e ir o rdin a r a rit
e a r, w e s a ll fi n d it
rat e o f in t e re s t t h e in t e re s t o n £ 100 fo r o n e
o re c o n e n ien t t o t a k e t h e in t e re s t o n £ 1 fo r o n e
ea r
s
m
a ll
hm
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teres t,
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Th e in te re s t
Pn r t at is ,
h
o
u
y
f P fo r
at
or
y
one
M
f
mu
ea r
is Pr,
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dt
h f
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P
:
n
F m( 1 )
P,
n
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r,
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n
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an
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t
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ree
h
i
v
g
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t
at
en
if o f t h e q
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u
m
m
v
m V th
L e t P b e t h e gi e n s u ,
t h e in t e re s t o f £ 1 fo r o n e
y
an
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f u h my b f u
231
To fi n d th e p res en t va lu e a n d dis co u
du e in a given ti e, a llo win g s i p le i n teres t
r
fo r
o ro n s
is ,
ro
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1
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A ls o
h
n
.
Le t P b e t h e p rin c ip a l in p o n ds , r t h e in t e re s t o f £ 1
b e r o f e a rs , I t h e in t e re s t , an d M t h e a o
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is
t
s
v
230
n ti
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n
t
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er
D t h e dis c o
o f e a rs
e,
y
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u
n
t,
200
HI
GH E R
AL
GE BR A
mu
.
fi y
a n d, s in c e
Th e a o n t o fP a t t h e e n d o f t h e rs t e a r is PR
t is is t h e p rin c ipa l fo r t h e s e c o n d y e a r, t h e a o n t a t t h e e n d o f
2
Si ila rl t h e a o n t a t t h e
t h e se co n d
e a r is PR x R o r PI3
3
e a r is PI3 , a n d s o o n ;
e n c e th e a
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e n d o f t h e t ird
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t at is ,
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h
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y
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y
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on
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re c k o n e d
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y
m
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;
PR
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1
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r
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r
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wh th im
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w imp l i t
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mu t f £ 1 i
mu t f P i 4 gy
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mu
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hv
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one
I n b s in e s s t ra n s a c t io n s
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s al t o
of a
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ear
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233
ra c t io n
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n
£ 1 fo r
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u
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mu
mp u
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e a r t e re is a
o n ce a
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dis t in c t io n b et e e n t h e n o in a l a n n u a l ra te o f in t e re s t an d t a t
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a
t
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y
if t h e in t e re s t is pa ab le t ic e a
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or
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m
hm
w
f in t e re s t , th e
h f
in
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th e
so
t
h
wh
o
a
le
mu
o
y
th e
at
h
hu
m
y
n
e ar
tru e
t
o
f y
f £ 1 in h a l
th e
a
an n
u
mu
o
al
n
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o
a
ear
Z
is 1
2
f £ 1 is
ra t e
o
f in t e re s t
r
,
1:
2 34
th e
n o
r
9
9
,
an
m
dt
u
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h
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c ase
es a
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ti
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y
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in al a n n a l rat e , t h e in t e re s t
.
y
.
th e
a
mu
o
n
t
t h e in t e re s t is
o
f P in
s a id
to b
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on
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e
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a
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co n
v
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ITI E S
2 01
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v y mm t
v u
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v
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To fi n d t h e a l e o f t h e
b e c o e s in n ite l gre a t
t en
1
r
s o t at
rx ; t
s
ut
,
q
p
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fi
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fi
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n
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en
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qi
in
s
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v
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ill in n
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e,
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o
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co
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pl
o
n
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c
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p
th e
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a
,
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t
a
u mb
e,
fi
D
P (1
=
er
n
hv
a
wd
1
%
b er o f y e ars ; t
10 g
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(
,
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en
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1 26
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67 2
25
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3
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.
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lo g 3 ,
4 lo g 2 lo g 3
2 — 5 10 g 2 — lo g 3
‘
.
hu
s
at
mi
th e ti
e
s ve ry n e a rly
n
4 1 y ears
7 2 7 00
01 7 7 3
.
m
4 1,
v
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t,
.
t h e p re s e n t
e
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3 01 03 , lo g 3
67 2
t
y
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p ze
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1 00
um
su
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:
——
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.
b e th e
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of
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p
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e re
n
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t t o P, w e
n
4
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e
—
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lo g 2
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mi
Th e pres en t al e o f £ 6 7 2 du e in a c e rtain ti
1
in tere s t at 4 7, p er c en t b e allo e , fi n d th e ti e ;
e
n
.
m
m
mu
o
A
rt 2 20,
[
it e
n
1
To fin d th e p res en t va lu e a n d dis co u n t
2 65
du e in a given ti e, a llo win g co p o u n d i n teres t
.
1
x
ru r
,
Pe
s in c e a:
a
,
hu
z
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t
.
en
o
e
hv
a
£126 ;
in g gi en
s
v
HI GHER
2 02
AL
E XA M PLE S
h
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d
e n re qu ire
lo g 2
d
v
i
g
Fi
t eres t ;
in
mu
rit
a
l
o
g g
en
o
a
a.
.
hm my b u
s
lo g 3
8 4 509 8 0,
lo g1 1
a
°
e
s ed
.
4 7 7 12 13,
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of A n n u ities a n d R evers i o n a/ry Pa y
en ts , a n d t h e a rt ic le A n n u ities
in t h e E n cy clo p aedia B ri ta n n ica
2 39
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S pp o s e t at a t e n an t b y p a in g do n a c e rt ain s u
244
e
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g
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fi nd t h e
XI X
C H A PTE R
IN E Q
u
.
U A L I TI E S
y
a is s a id
A N Y q a n t it
245
—
is p o s it i e ;
n
a
b
e
i
n
t
6
a t
q
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u y wh
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en o
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t ra r
d po s
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.
24 6
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.
b, t
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en
it is
a
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I n t h e p re se n t
is dire c tl s t at e
a n t it ie s
q
y
h
b e gre at e r t an an o t e r
—3
t
s 2 is
re a t e r t an
g
,
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— 5 is le s
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to
v hu
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v
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.
a
b
c
c
i n equ a li ty will s till h o ld after ea ch
in crea s ed, di in is h ed,
u lti lied, o r divided b y th e
p
u a n tit
q
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t
is ,
at
an
m
m
.
24 7
by
a
If
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6
a
to
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c
s ide ,
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a> )
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c
s
fro mo n e
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s ide
at
to th e
b, t
h
en
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in
an
o
th er
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;
mmy
in equ a lity an y ter
h
an
e
i
s
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n
c
d
t
s
b
e
f
i
g
g
a
b e tra n sp o s ed
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y
tl
b
a
;
is, if th e sides of a n i n equ a lity b e tra n sp o s ed, th e
u s t b e revers ed
in equ a li ty
t
at
m
.
s ign
f
o
IN E Q
If
h
b, t
b ) is
a
a
U A LITI E S
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n e a t i e , a n d t e re
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en
v
h f
th e
of a ll th e term
s of
if
s ign of i n equ a lity m
u s t b e reversed
b, t
h
en
b,
a
an
if
th e
If
dt
b
a, >
a
I
24 9
If
.
1
o r a
b,
a >
1
b
”
d t
a n t it
q
?
an
v u
Fu h
o s it i
e
rt
2
e re o re
g
g
7
u
q
h
t
5
a re
t
an
h
o
f
e
z e ro
v
at
th a n
y
h
v
is ,
at
h
wh
in t e ge rs , t
e
b
a
"
,
2
hus
T
j
— 2a b
2
b
y
.
en
is
e re n
an
y
f
o
(
b)
a
+
2
b
2
>
2a b
2
t it
y
is p o s it i
is p o s it i e ;
an
v
v
e,
an
d
0;
.
zz”
Ja
)
m m
m m
u
y
m
l
i
t
b
q
u
q
two p o s i ti ve qu
an
ti ties i s grea ter
.
ec o
a
es
u
an
e
l
w
u
i
t
y
h
q
a
en
th e q
w
u
2 51
Th e res lt s o f t h e p re c e din g a rt ic le ill b e
l, e s pe c ia ll in t h e c a s e o f in e q alit ie s in w ic
in o l e d s
e t ric a ll
u fu
e
b
re a l
A
is , th e a ri th eti c
ea n
th eir geo
etric
ea n
Th e in e
e u aL
q
a re
t
e ry
.
a:
l
ar
b
7
m
it
b
>
m
sa
at 1 s a
( 77
mil
p
e
a
o s it i
1)
"
a
a
Si
a re
p
us
b l bz b 3
m
e
.
n
re a t e r
e re o re
s
e re o re
1
s
2
3
by
reversed
u
b
>
a
a
h f
<
a
b
>
d if p , q
an
1
Th e
.
h f
y
e r,
2 50
se
e
h f
mltip li d
mtb
s
d a a
h
v
i n equ a li ty b e
an
ides of a n i n e qu a li ty b e
u a n tit , th e s i n o
i
u
l
i
t
n
e
a
q
y
g
f
q
y
t
g
i
be
ac
t
at
.
A ga in , if a
p
n e
o re
s ign s
th e
en ce,
d b — a is
b
a
h
an
2 09
.
.
y
y mm
v v
H
.
H
.
A
.
y
u
.
h h
an
t itie s
fu
o
n
th e
a re
v y
d er
le tte rs
210
G
H I GH ER
m
E x a p le 1
I f a, b ,
.
d
c
A L E BR A
v u
te p o s iti
en o
e
q
an
.
titie s , p ro
v h
e
t
at
2
2
b
c
2
a
wh
by
en c e
my b
It
dd
a
a
itio n
e no
A gain , fro
a
tice
m( 1)
2
t is
at
t
a
2
b
2
b
dh h
c
C
re s
b
‘
u
2
2b c
2
26 a ;
2
2a b
2
bc
u
lt is tr
be
c
3
ab .
ca
2
fo r
e
w
ritin
dw
n
o
g
si
ar in eq
h d
h d
o ul
s
fac to r b + c ,
lo n ger o l
u
c
al
es o
f a, b ,
0
.
)
alitie s an
( b + c ) + ca
2
It
3
mil
th e two
v u
real
be
b + c > b c (b +
By
y
an
d
a
dd
i n g, we
(c + a) + ab (a + b )
Ob
.
d m
v
u
vd h
h
h
tain
du
2
n tro
i
n g th e
c
b e Ob s er e t at ( 3) is Ob tain e fro
b
i
)
y
(
ill n o
an d t at if t is fac to r b e n egati e th e in e q ality ( 3 )
w
.
m
E x a p le 2
2
3
x + 1 or x +x
.
If
my
a:
hv
a e
a
3
( 477
1
v h
N o w (a:
is p o siti
e,
in g
as at +
I f a:
v
1 i s p o s iti
1 , th e in eq
u
3
we
.
h
h
fin d
wh h
ic
hv
a
v
H
en
33
or
b eco
x
2
+x
v h
u
m
; t at is ,
e s an e q
ality
d
acc o r
in g a s
at
S
is ,
(
b)
a
(
2
a
.
.
v
h f m
1
or
a:
S th
u
y
b)
e ir s u
m
2
,
e
S
2
-
if
is gi
is le as t
en
e n ce ,
,
is th e greater,
)
+
e o r n egati e
ality
u
4P = S
t
2
+ 1
4 ab
i
g
al e
L e t a an d b b e t wo p o siti e q a n t it ie s ,
d P t eir p ro d c t ; t en ro
t h e ide n t it
2 52
an
v u
en c e
x
d
real
y
.
x
ac co r
an
if
th e
su
wh
—b
(
a
v
en
,
)
2
,
an
d S
o
i s grea tes t wh en th ey a re
u a n ti ti es i s
i
n , t h eir
v
e
q
g
:
P is gre a t e s t
i
t
i
ve
o
s
p
equ a l
su
mi
s
4P +
wh
(
a
en
a =
b
an
d if P is
b;
a =
mof tw
2
t
i
i
t
es
i
s
i
v
en
g
q
an d
if th e p ro du ct
u an
lea s t wh en th ey
,
th eir p ro du c t
f two
o
a re e u a l.
q
p
os
itive
21 2
HI
2 54
is
t;
g
ex
p
s
t s, t h e
u
ct
m
f
o
t
H
.
en c e a
m
b
n
c
n
.
b
a
m n
l
a re a l
e
u
q
a
l, t
h
wh
is ,
at
a
m
hu
T
t h e gre ate s t
s
m
mn
m
E x a p le
fx
o
n
um
v
erically
Th e gi
th e
h
su
en c e
Fin
.
O
hu
T
x
s
h
le s s t
) (
3
e
a
fac to rs
x
)
‘
th e greates t
b
b
c
n
p
v lu
a
p
w ill
re s s io n
p
an
th e
en
h
B u t t is la st
.
.
wh
mi
f
d th
f t
a c t o rs
+ c +
wh
c
o se
s
su
e re o re c o n
a c o rs
c
3
p
7
en
e
is
”
v u
th e greates t
an a
en e x pres sio n
m f th
(a
d
n
ex
p
+
b
a
.
f
will b e gre a t e s t
”
wh en
.
is gre at e s t
o r a
s ta n
.
en
is t h e p ro d
re s io n
c o n s t an
a re
,
wh
re a t e s t
e
n
G E BR A
d th e grea tes t va lu e of
n ,
b ein gp o s itive in tegers
n
,
,
AL
" "
m
a b c
m
m
c o n s ta n
Sin c e
b
fi
To
.
GH E R
al e o
(
a
+
x
) (
3
a
—x
)
4
fo r
an
y
real
v u
al e
.
is gre ate s t
Of
f
h
t is
is greates t
v u
al e I S
wh
61,
1s
en
ex p ress io n
wh
63
8
m
m my
— (D
greates t ; b u t
i
or
2a ;
en
4
m mm m mmv u
u
u
f
mh
an d
al es
Th e de t e r in a tio n o f
ax i
u
in i u
a
l
h
O te n b e
re s i
e ff
e c ted b
t
e s o l t io n o f a q a d
o
y
p
y
ra t io
t
h
o re o in
d
s
I
n s t an c e s
of
b
e
a t io n
t
an
e
e
t
o
g
q
y
g
t is
a e
a lre a d
o c c rre d in
C ap 1 x ; w e a dd a u rth e r
ill s t ra t io n
m
2 55
f
.
u
h hv
u
m
E x a p le
is
a
h
y
D
en o
th e p ro
.
f
.
.
Div d
m imum
ax
h
u
.
.
i
e an Odd
in teger in to two in tegral p arts
wh
o se
p ro
du
ct
.
te th e in te ger b y 2 n + 1 ;
c t b y y ; t en ( 2 n + 1 ) x
du
h
th e two p arts b y
2=
en c e
x
y ;
wh
x
an
d 2n + 1 —x ;
an
d
b u t th e q
u
h
b e greater t
v u mu
al e
an
st
d 71 + 1
2 56
ud
tity
1
an
an
n
d
mu t b
ic al
ra
s
be
71
So
mtim
.
Fin
4
wh h
ic
; in
71
-
d
an
.
m
e
d
h
ex p res s io n
my
we
es
Pu t e + x = y ; t
H en c e th e
hu
an
d th e
s
mi imumv lu
th e
a
n
.
2
.
3
.
(a
y) (b
C
is
e x p re s s io n
mi imumv
th e
n
c o rres p o n
4
,
5
.
ce
e
ro
e
t
u
al e
a
2
(
at
b
(
at
+b
c
th e
les s t
2=
l,
an
su
is
r
/( a
e
d
a
th e
en
o
2
x
2
an
) (b
e
11
.
are n
5,
s
h w th
a
e
+y
x
s
3
e
t
at
ro
e
t
at
t
.
at
a
2
z
a
2 2
0
t
e
2=
4 a b ay
a
8abc
u
are
-
C
mi
te r
b
H
s z ero
— C
y
.
an
tity
s
hw h
t
e
at
b
a
2
2 x
)(
a
a
cit
b
)
2 2
b
b
y
“
,
an
z
+ by < 1
hw h
t
e
at
iii
)
d
y
?
e
2x
or a
3
2
)
2b
3
.
4
.
ea
a
.
;
2
3a b
4
c
b
ax
(
bc
c
(a
a
)
a
b
b
c
)
.
(
a
b)
.
‘
h
.
i
o
s
i
t
e
p
)
; t at
.
v qu
real
l,
b
e
2 2
0 a
a
s
rea te r
3
a
c
sq
/(
a
‘
d
at a
2
c
ro
e
d
o
.
h w th t ( y y
F i d wh i h i h g
h b b
P v
h 6 b b (b
P v
Sh w h
b
e
e
)
.
b)
) (a
mf
an
a
mg mt h
—6 +b —c+2
s
XI X
.
by )
ax
a
an
If
n
.
th e two p arts
s
t
gre ate s t
is
) (c
h
+
n
m
y) (
ab
at
er
s n e
If
S
10
c an n o
f
e o
mi imumwh
a
al e O f x
in g
t
e
If
.
hu
y
ral 1 te
e
; t
or n
ax
9
mt g
18
y
e re fo re
y)
e
d v u
v h
P v th
Sh w h
v
li
Pro
re c i ro ca
p
m
h
dt
e , an
en
E XA M PLE S
1
v
t h e fo llo w
u se
a
N/y
T
s
= n + 1,
cas e x
213
.
p o s iti
e
1
or
4
2
UA L I TI E S
.
E x a p le
th e
th e
er
INE Q
an
d
21 4
HI
.
13
t
15
a
19
3
3
reate r x
e
a
x
e
re ate s t
t
a
a
x
f
e
o
3
hw h
20
.
S
21
.
S h e w th at
at
x
a
t
o r er
f x2
1 2 x + 4 0,
t
3
2 7 xy z
( |Z)
t
(
an
23
Fi
2
in teger greater t
e
*
2 57
m
m
a + b
.
d
m mu mv
d
th e
mi imu mv lu
ax i
u
al
e o
a
n
To p ro ve tha t
hv
b i
a
e
s
ex
a
be
m imum
ax
h
2,
an
s
hw h
t
e
at
f (7
x
)
)
.
.
4
) (2 + x )
5
wh
w
b et
en x
ee n
.
2
a
d th e
my
"
n
3
ex c ep
We
+ 11
V
1 +n
th e
n
.
x
3
—z
n
2
fx?
es o
+
n
—z
Fi
a
,
2
( )
.
7
at
1
( )
22
7 an d
e
.
d h
in
x
if
,
d 2 4
an
,
+z)
1
”
os
n
”
2”
t
3
e o
a
2
e a
e
v v lu
+ x + 2 fo r p o s iti
.
at n
n
.
a
.
2
9x
e
.
2
5ax + 9 a
n
e
t
e
.
GE BR A
+
t
8
e
AL
or x
2
1 3a x
3
h w h t ( In)
Sh w h t (
y
Sh w th
i
If
b
i
v
p
S
.
18
x
f 24 x
e o
17
t
n
.
16
t
an
t
s
n
.
r
t
e
r
ea
g
v lu
c
e
.
14
hi h i h g
Sh w h t
Fi d h g
v lu
h
Fi d h m
i im
umv lu
W
12
GH E R
re s s io n s
p
in
a
+
b
le s s t
a s c en
t
if
e 0f
a
an
wh en
1
d b
m
is a
x
a re
i
o
v
an
s
t
i
e
p
o s i ti ve
p
d
u ne u a
q
p rop er fra c ti o n
l,
.
'
"
h
an
2
din g p o
w
e rs o
m( m
l
my
we
2
1)
a
f
2
b
a
2
ex
p
an
A
rt
[
d
e ac
h
o
f t
h
e se
21 6
HI
GH E R
AL
G E BR A
.
u
u
u
u
H e n c e s o lo n g a s a n y t wo o f t h e q a n t ities a , b ,
a re
n e
al
q
“
”
"
"
16 c a n b e di in is e d wit h o t
c
b
t h e e x p re s s io n a
an d t e re o re t h e
alt e rin
h
a l e Of a
c
h
t
e
b
al e
g
“
"
"
'
”
76 will b e le a t w e n all t h e q a n t it ie
Of a
c
b
al
a, b,
a re e
I
n t i
c a se ea c
Of t h e
an t it ies is e
al
q
q
q
’
’
v u
u
’
m h
s
’
’
hs
.
h f
h
h
v u
u
s
u
u
n
an
d th e
H
v lu
a
e Of a
wh
en ce
en
"
b
’
"
’
’
c
b,
a,
"
h
u
a re
n e
"
’
u
q
t
h
en
m
b eco
es
a l,
717
m
m m
u m
u
v
m
v
i
ro
If
l
ar
an n e r
lie s b e twe en 0 a n d 1 we
a
i
n a si
p
y
t at t h e s ign o f in e q alit in t h e ab o e re s lt
s t b e re e rs e d
h
u y
my b t t d v
i th mti m
m
t
h
p
f
th
t
h
i
th m p w
f
Th e p ro po s it io n
Th e
ar
e
is grea ter
ex c ep t wh en
*
p
2 59
o si tive
mli
If
.
u an
q
s
s e rie
s e rie
f
s
A te r
c o rre s o n
p
th
e
o
db
a re
p
o wers o
e r a ri th
er o
o
an
d 1
1
co n
p
o s i tive
mti m
e
c
ean
u an
ti ti es
l
c a s es
q
al
in
an
d
a>
b,
an
d
x be
f
i
a
1
+ x +
sisti
s
m;
f
a
g
1 t er
o
—
+ x +
n
l
6
g fb
o
+
i
9
s
an
ln
eco n
er
an
ed
.
e
d
2
I
E
1
+
m
1 ter
s
.
m
h
m
t
d gt m
m v
th
th
um
m
he
h
n
ws
ti ty ,
c o n s is t in
is gre a t e r
is e s t a b lis
o llo
.
.
in tegers ,
o si tive
as
f
th
e
b etween 0
es
a an
b
th e
ea n
y f
e rb all
e s a e
a
c
an
1
th e
v
e
h
d t er , eac t e r
re a t e r t a n t h e
1
i
s
of
g
( )
o f
2)
b e r o f t e r s in ( 1 )
o re o e r t h e n
n
b e r o f t e r s in ( 2 )
e n c e t h e p ro p o s it io n
um
h
m
l N E Q U A L ITIE S
*
26 0
To p ro ve th a t
.
if x
an
1
d
y
a re
p
ro
p
er
f
ra c tio n s a n
as
1
+
y
—
*
hu
26 1
dedu
ce
s
t h e p ro p o s it io n is
.
th a t
a
1
(
1
+
2
1
+
en ce
x
y
.
1
+ x
1—
x
p
v
ro
y
ed
x
)
.
—x
l
(
)
b
’s‘
>
1,
b
1
(
by P; t
h
en
(C
H
d
<
2
To p ro ve th a t
D en o te
an
2
l
dt
’
o r
1
an
y
d p o s i tive,
> OI
din g
+
l —y
’
a cc o r
217
.
lo g P is p o s it i
1
(
v
+
e, an
dt
h f
e re o re
P> 1
1
.
if
x
<
l,
an
d to
218
H I
h
I n t is
re s
ul
t pu t
o
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2 2 44
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V A N I SH I N G
27 1
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a
b
'
b
0, t h e n
m u pp
2 3
i
u
q
s
at o n
e
a
-
b e co
m
is
t
i
n o
o se
e
u
q
;
If
a re n o
H
C
E
is
al
c an n o t
e
u
q
is , t h e
ro o
fi
o
c o e f c ie n
if t h e
re a t
at
t
f
f
t
o
x
is
al
b
u
’
c
ca
y
b
a
s
a tio n s
'
'
a: an
to
d y
ub
'
s t it
'
’
ab
c a
'—
a
'
b
b o t h in
a re
u ti
n
t h e t wo
c,
y i th
ti fi d b y
l
on
be
sa
c,
we
to
eq
fi
fo r
g
it e
n
a
’
h
I n t is
.
b
,
’
th e
,
c a se
d
s ec o n
by
0 diffe r
in co n s is ten t
b
'
by
es ax
I
If
O
h
.
bc
’
t
o r 00
yg
y mll
.
es
is in de fi n it e l
a t io n
a
m
b eco
90
en
n
e
s
hv
a
e
u
q
e ir
a tio n s
ab s o
fi
y
an
n
v
a
u
l
s,
es o
an
f a:
d t h e t wo
an
z0
c
m
u
l te te r
it e
e
by
ax
d
an
d b e in g
dy
an
u
eq
.
s
a tio n
w ide n t ic a l
.
e re ,
si
n ce
bc
’
b
’
0
c
an
d
'
’
ca
O th e
c a
u
va l e
s
o
f
x
an
dy
s luti i i d t mi
I
w
lly
ly
hv
u
i
t
q
i
d u h
u
t
m
b
s
i
fi
t
d
y
q
it d u m
b
f v lu s
A
u lim
t
by
[
i
w
i
l
i
u
t
y
y
Th
d
h
A
t
l
G
m
t
d wh i
will
q
h v diffi u l y i i t p i g t h
u lt i
ti
wi h
t igh t li
t h g mt y f h
u m th f m6"
f t i th p s t
w
i v lv i g w u k
e ac
ac
h
o
n
e
o
re a
e n o
eo
t
o
t
an
d th e
n s,
o
s ac
n
n
a
s
an
c
e
a
a n
n e
e
an
e
r
.
n
on
rea
on
a
e er
n a te
on e
e
e
a
n
.
a
a
s
on
e
.
na
es e re s
.
s
on
t
e
e r re t n
e s ra
o
e
c a se
er o
n
er
c
e r
n o
n
e
n
e
a
t
n
an
re e n
e
n
or
e
e
as s
,
0
ca
s
n
eo
co n n ec
e r
on
t
HIGHER
2 28
AL
G E BR A
E XA M PLE S
XX
.
Fi
n
d
mit
t h e li
wh
1
( )
(2x
7x
2
(
1
2
Fin
e x p re s s io n s ,
h
w
2
( )
,
( 3x
2
— 6 a'+ 4
5
3
x
—
3
(
2x
d th limi
ts
gti
wh en
_ e
o
_x
2
2
x
wh
en x
—
wh
,
wh
1:322i ;
en
x =
x =
en
=
en
g
1
o
2a
.
.
.
‘
1
(
a
(a
3
(a
e
>+ (
2
x
ca— x
)
x
wh
17
.
n
)
)(
3
—
2
( x D
(3
)
m
+ 1)
-
-
wh
lo g
wh
en
i
.
'
wh
en a =
en
wh m
— 2x
1
(
)
2
x
'
—
2
5a
7x)
o
.
wh m
O
2a
+
lo g
3x
Jo — 3
,
x
2
+9
(7x
?
:
5 22
Jx
.
f
e
,
4
1—
7
O
2
(
?
x =
en
cs — 5
—I
e
.
)(
)
—9 1 + x
)(
)
x
x
n
— 5x
3
)
(
)
x
2
3
+
(
3
4 56
wi g
f t h e fo llo
x = oo
en
—3
s o
.
en
O
.
en
n = oo
n = a;
.
_
a
.
CH A PTE R
R GE N C Y A N D
DV
I
XX I
ER
s
.
GE N C Y
O F SE R I E S
s
.
mi t
u mb
is c a lle d a erie ; if t h e s e ries t e r
it is c a lle d a fi n it e s eri e ; if t h e n
it is c a lle d a n in fi n it e s erie
we
s
h
s
u u lly
al
l
s
+ u
u
hv
a
s
n a
e at
er o
f
.
de n o t e
a
s e rie s
by
+
s
fu
m
s
S pp o se t h a t w e a e a e rie c o n s is t in g o f n te r s
e ries will b e a
n c t io n
o f n ; if n
in c re a s e s
Th e s u
o f th e
e it e r t e n ds t o b e c o
e e q a l t o a c e rt a in
in de n it el , t h e s u
n it e li
i t, o r e ls e it b e c o e in n it e l gre at
27 7
fi
fi
m
.
s
y
m
fi
m h
ms fi
s
.
m u
y
.
wh
A n in n it e s e rie is s a id t o b e co n v ergen t
e x c ee d
e ric a ll
rs t n
ter s c an n o t n
o f th e
a
b
a n t it
w
re a t n
e
h
o
e er
y
q
g
fi
y
u
A
th e
u
q
fi
an
in
n
rs t n
t it
27 8
m
v
fi
n
it e
m
ter
y by t
s c an
a kin
be
s
s a id
y
uffi i
t o b e div ergen t
e n
c en
su
e
n
smfi
o
m
it e
.
md u m i
a
th e
yg
tl
yg
e r c a ll
re a t
wh
re a te r
en
t
h
th e
an
su
an
mf
fi
y
o
n
ite
.
m
fi
ms
v
fi n d th e s u
rs t n t e r
o f th e
o f a gi e n
s e rie s , we
a
as c e rt a in
e t e r it is c o n
e r e n t o r di e rge n t
y
g
b y e x a in in g
re
a in s
n it e , o r b e c o
e s in
e t e r t h e s e rie
n it e ,
e n n is
a de in de n it e l
re a t
g
.
I f we
g
n
m
is
s e rie s
um
en
m
c an
wh h
v
m wh h
s m fi
m
fi wh
fi y
F
mpl t h m f t h fi t t m
m
.
o r ex a
e,
e su
o
e
rs
n
er
s o
"
l —x
'
—
l
a
v
f th e
s e rie s
230
If
fi
n
is
a:
n
mit
x
be
di
v
e rge n
n
u m i lly
er c a
an
s
1
re a t e r
t
g
t
AL
1 , th e
an
s e ries
is t
re a t e r
t
h
g
h
fi
y
an
n
v
I f a;
f
an
b e twee n
ic h
m f th fi
o
s
e
e ri e s
mf
b
dd u m
o
su
n
e re o re c o n
1 , th e
an
s
p
v
su
m
c en
r
g
an
mi
rs t n
ter
s
b
eco
s n
,
m
es
—1
a
e
v
e
an
u mb
ms i
en
+
ms i
f ter
er o
n
f t er
s 1
T
al e s 0 a n d 1
c alle d o scilla tin g o r
e
ri o
p
er o
v
th e
my b
wh
a
.
1 , th e
o
m
h f
n
it e
1
o
su
u ffi i tly
i
u
y
t
t
;
q
d b y t a kin g
an
.
.
If
th e su
s e rie s is di e r e n t
g
Th e
GE BR A
a:
a
o
e
ess
d th e
an
1
is
md
a
er c a
1
it e li
If
HIGHER
u m i lly l t h
u
.
my
w
an
c a s e s in
27 9
Th e re a re
h
rs t n t e r
h
f
t
e
s
n din
t
su
o
o f
e
g
b y wh i
t e re o re t o in e s tigat e r le
i
of a
en
se
o r di e rge n c
e rge n c
g
.
fi
h f
v
y
s
u mmt io
a
v
v
n
m
m
fi
u s
y
v
.
m
fi
A n i n n i te s eries i n wh ich th e ter s
2 80
eac h ter
t
i
v
a n d n ega tive i s c o n vergen t i
o
s
i
e
f
p
less tha n the p recedin g ter
.
mi
m
a ltern a tely
a re
n u
s
mi
er c a ll
y
.
s
Let th e
e rie s
u
wh
e
de n o t e d b y
u
-
l
+ u
,
f m
v
s
en
u
4
5
w itt
be
a
r
in
en
eac
h
o
f th e
f ll wi
o
o
n
g
s :
(
)
u
F m( 1 )
i
i
t
v
u
o
e
p
q
ro
o
3
my
e rie s
u
a
u
e re
Th e gi
or
b
s
ms i
f te r
we
an
s
t it
le ss t
h
t
s ee
y;
an
an
u
s
h
a
s
)
(
at
d
u
u
4
th e
,
(
)
u
su
u
5
(
)
5
m
o
en c e
e
s
an
n
u mb f t m i
m f y umb
v g t
y
t at t h e s u
s e rie s is c o n
f m( 2) h
h th
ro
f
a
er o
o
an
er e n
.
er
n
s
s
er
HIGHER
232
fi
AL
G E BR A
.
m
m
m
a n d a ter s o
e
A n i n n ite s eri es is c o n vergen t if fro
f
is n u eric a lly
to th e p recedin g ter
th e ra ti o of ea ch ter
x d ter
les s th a n s o e qu a n tity which is i ts elf n u eri ca lly les s th a n u n i ty
2 84
fi
.
e
m
m
L e t th e
h
T
m
t
ro
“
<
h
at
t h e gi
en c e
e
de n o t e d b y
v
3
26
26
I
]
1
1
s e rie s
en
26
.
is
co n
v
e r en
g
t
.
i
'
I n t h e e n u n c ia t io n o f t h e p re c e din g a rt ic le t h e s t u de n t
285
“
ro
o rds
a n d a te r a
o ld n o t ic e t h e s ign i c an c e o f t h e
”
x e d te r
.
hu
fi
s
u
s in c e r
is ,
H
te r
en
it
t
x ed
e
.
mb
f mh fi
b egin n in g
s e rie s
m
m
fi
m
f m
w
‘
f
.
C o n s ide r t h e
s e rie s
+ na
H
1
n a;
e re
n
ufi
— l
n
— l
m
y
h
d b y t akin g n s f c ie n t l la rge w e c an
a k e t is ra t io a
p
ro x i
a t e t o x a s n e a rl
w
l
a s e , a n d t h e ra t io o f e ac h t e r
a
s
e
e
p
p
t o t h e p re c e din g t e r
will lti a t e l b e x
H e n c e if x < 1 t h e
s e rie s is c o n v e r e n t
g
an
m
y
m
u m
y
m
.
.
Bu t th e
t
h
at
is,
u
n
t il
w ill
ra t io
n
we h a
1
v
n o
t b e le s s t
h
an
1,
u
n
t il
x
v
m
wh h
ic t h e t e r s
f a c o n e rge n t se rie s in
n c re a s e u
t
i
o de c re a s e
a
t
a c e rt a in p o in t a n d t e n b egin
o
y
p
99
Fo r e x a ple , if x
t en — l
1 00, a n d t h e t e r s do n o t
1
x
1 00
b egin t o dec reas e n t il a t e r t h e l o 0 t e r
m
H
e re
e a c as e o
m
h
u
f
h
m m
.
.
m
“
D
C O N VE R G E N G Y A N D
GE N C Y
IVER
O F S ER I E S
233
.
m
m
A n infin i te s eri es in whic h a ll th e ter s a re Q/ th e s a e
28 6
a n d after s o
s ign i s divergen t iffro
e
x ed ter
th e ra tio of ea c h
is grea ter th an u n i ty , o r e qu a l to u n i ty
ter
to the p recedi n g ter
m
m
mb d
.
m
Let th e
u iy
t
n
o
ea c
h
m
,
f n te r
s
I f th e
x e d ter
fi
th
fi
u
u
mi
.
h
h
th
is gre a t e r
reat e r
g
s
th e
en c e
t
h
ms
u
u
s e rie s
is
y
n
v
.
s
mit
t h e li
o
f
b e de n o te d b y
m
o
f t
fi
n
e r en
g
t
a
h
er
is
,
wh i h
v
c
co n
yi
it e l
t e s ts,
ese
n c re a s e
is
If A
1 , th e
s e rie s
is di
fu h
v
co n
v
t
e r en
g
m
w
u a lly
tin
co n
—l
defin i tely
a n t it
r
q
u
rt
2
A
8
4
[
.
t in
.
A rt
y pp
28 6
a
h
pp
en
v
id
is
o
ht m
er
ie n t t o fi n d
mit
h
d ; le t t is li
]
v
ro a ch in
h
.
hi
n
1
g to
o 1
t
g
f
as
a
ils
I f,
.
mit
it s li
as
e
t,
n a
an
e
26
e r,
a
n
en
it
—I
h
v
is di
s
ni
an
1 but
>
s e rie s
th e
e rge n
en
s
n
,
a
d y e t gre a te r t
hwv
o
v
m h pp t h t
it wh
i t lim
i
m y fi t
t
can n o
an
di
o r
er en
g
e
A
.
co n
t by
.
.
We
s
h
a
ll
u se
“
m
Li
as
n
mit
t h e li
f
o
n
m
E x a p le 1
ergen
H
u
f A rt 2 8 4
o
ro a c
a
f
is
c
t h e te st
u a ll
eac
a
A
rt
[
.
I n t i s c as e we
its e l le ss t h a n 1
i n creas ed
y wh i h
en c e
to
A
s e rie s
1 bu t
f
1
-
1 , th e
in
v
m
m
f t h e t e r s a t e r th e
o f n t e r s is
re a t e r
g
o
m ft
m it
is in de
n
If A
u
H
”
en
.
a
b
e e it e r c o n
er en t
I f A = l , t h e s e rie s
y
g
a
ill b e re q ire d 5 fo r it
d a
rt e r t e s t
y
an
n
wh
"
h
v
m
.
u
‘
u
it
u
u
u
it , e a c
an d th e su
di e rge n t
an
an
°
I n t h e p ra c t ic a l a pp lic at io n
287
a in
t
o a sc e rt a in t h e p a rt ic la r t e r
g
h
re c e din
t
r
an t h e
re a t e r o r les
t
e
p
g
g
hv
m
d ter
e
e n o te d b
“
h
I
f
t
ra t io is e
e
a
l
t
o
y 1
q
o f t h e s c c ee din g t e r
is e q a l t o “ n a n d t h e s u
is e q a l to n u
e n c e t h e s e ries is di e r e n t
g
xe
ra t io
a n n u
mfi
t
ere
o r
di
.
—l
Fin
v
erge n
u“
t
wh
en
d wh h
et
n
er
ab
an
re v ia t io n
b
—l
is in
th e
fi
n
it e
s erie s
o
f th e
w
o rds
.
wh
o se n
‘h
mi
ter
is
s
.
(
n
+ 1)
u
“
3
a
co n
HI HER
23 4
h
en c e
h
If x : 1 , t
m
E x a p le 2
v
erge n
co n
t
en
Li
div
or
s erie s
is
co n
if x > 1 th e
s eries
is
dv
mJ
“it
1,
1
2
2
f
2
?
rt
If x
i
ergen
er
tes t is
2
3
x
x
m
pl
e
3
1)
is
co n
if x > 1 th e
s erie s
is
div
I n th e
.
n
s e rie s
b ec o
seri es
(
n —l
1 th e
if x
1 th e
m
1
es
2
2
2
3
2
m
un
u
s
if r < 1 th e
s e rie s
.
re q
u d
ire
.
x
v
ergen
ergen
2
4
2
t
t;
.
an
d is
o
v u dv
b io
i
s ly
erge n
t
.
s erie s
Li
hu
t
_2
n
—l
n
t
t;
erge n
J
e n ce
xa
u h
a
v
t?
erge n
u
E
d
an
.
s erie s
I s th e
.
GE B R A
if x < l th e
H ere
H
AL
G
is
n
-
Li
+
(
1) d
n —
l
v
co n
m
a
e rgen
t,
an
d)
d th e
su
mi
s
fi n ite
.
[ See A rt
.
fi
6 0, Co r ]
.
If there a re two i n n i te s eri es in ea ch of wh ic h a ll th e
ter s a re p o si tive, a n d if th e ra ti o of th e co rresp o n di n g ter s in
n i te, the two s eries a re b o th co n ver en t,
th e two s eries is a lwa y s
g
o r b o th divergen t
2 88
m
.
m
fi
.
L e t t h e t w o in
fi
s e rie s
it e
n
b e de n o t e d b y
v + v + v + v
3
4
2
l
v lu
Th e
a
e o
f
f th e
u
ra c t io n
+ u
,
e en
t h e gre a t e st
“
v
an
d is t
h f
e re o re a
u
H
s e ries
p
ro
o
p
st
2
n
+
l
J
3
i io n
fi
.
dl
o
“
u
E
v
v"
s
t it
+
v u
y
,
L
sa
y
l
- -
v
2
fi
ra c tio n s
‘
,
an
f
f th e
E
n
i te q
u
an
if o n e s e ries is
in n ite in
al e,
en c e
is
,
+ u
fi
3
2
l
w
2
v
v
lie s b e t
+ u
u
it e in v al e ,
s o is t h e o t
n
h
+ v3 +
is t h e
ic
e r;
so
+ v
n
o
wh h
t
h
)
.
er
p
ro
v
if
es
HIGHER
236
E xa
m
pl
Pro
e.
v h
t
e
th e
at
is di ve rgen t
Co
a
u
.
mp
are
th e gi
hu
T
s if a n
xiliary s eri es
v
d
vn
v
d
1
te th e
th
ter
Wi t
en s e ri e s
an
h
1
en o
Li
di
m
u ,,
_
1,
vn
v
erge n
t
v
.
is di
s erie s
h
T is
co
291
mit
li
o
B u t th e
ergen
t
a
u
h
m
u
th e
e es
I n th e
+1
1
2
a
so l
dv
i
erge n
e
1
h
bot
serie s are
is
s eri e s
s erie s
en
n
th e two
tio n
mp l
f E xa
o
li
c a t io n
pp
o
s
v
f th e gi
o
+ 1
n
n
t, t
h
co n
v
t
als o
th e gi
erefo re
or
h
bot
ergen
v
en
e
.
y th
n e c e s sa r
.
e
n
A rt 2 8 7
.
f A rt 2 8 8 it is
o
it ; t h is
h
u
ld b
fi
i
h
ll
w
m
w
s
f
g
y
fi
s
a e
ere fo re
x iliary
1
.
mpl t
.
dt
an
n
hv
we
re s p ec ti ely ,
71
en c e
.
4
u
h
GE B R A
s eri e s
3
2
AL
W ill b e t h e
c a se
th e
at
if w e fi n d
our
n
au x fli a r
Ta k e
u
th e
“,
t
in
s e ri e
n
th
h igh t p w f
i fi i
by A t
Z
l
y
th
es
”
s
te
n
n
e rs o
o
r
o
e
o
m f th
ter
o
i
g
e
D e n o te t h e
.
2 7 0,
.
a
an
d
v
n
y
v
s e rie s
en
re s
my
u lt
by
an
v
n
b e t ak e n
a
d
; t
as
h
re ta in
en
th e
th e
n
th
y
lim
it f
t m f
on
l
th e
o
o
er
n
s e ri e s
e a u x i i ar
m
E xa p le 1
dv
i
ergen
As
n
t
.
S
.
hw h
t
e
th e
at
s e rie s
wh
th
o se n
mi
ter
s
f/
3n 3 + 2 n + 5
.
in c re as e s ,
a
n
mt
ap p ro xi
a es
v u
t o th e
z/es
al e
‘
H
h
t
dv
.
e rgen
s erie s
h
B u t t is
t
.
we
i f vn
th e
erefo re
serie s
i
e n ce ,
wh
se rie s
w
hv
a e
o se
is
th
n
dv
i
or
Li
m
mi
t er
ergen
w
1
w h
hi c
s
"
1
?
my
a
12
72
t
[A rt
.
t
h
is
k
b e ta
erefo re
a
en
fi n i te q
as
th e
v
th e gi
en
u
an
u
a
ti ty ;
i
xiliary
s erie s
is
CO
E xa
NV
m
pl
2
e
GE N C Y
ER
Fin
.
A N D E I V E R GE N C Y o r S E R I E S
d wh h
et
3
is
v
co n
t
erge n
o r
dv
i
u
e rge n
t
J
"
ii
s
wh h
ic
in
s erie s
th e
er
+ 1
237
.
71
.
.
1
L6
_
9a
1
1
2
9n
1
1
3n
k
I f we ta
B u t th e
u
a
hv
we
e vn
a
e
x iliary s erie s
1
—
‘
1
is
co n
v
292
mi
Le t
u
t
h
,
2
2
v
th e gi
re re s e n
p
+ 1
"
2
3
23
en serie s
1
x
t th e
ergen
f
1
(
(
r +
o
m
d
—r
1
t
.
+ x
)
b y th e B in
n
.
r
an
”
n
,
+
x
u
h
v
co n
m
ter
s
o
N ow
s in c e
sa
m
r
en
x
e Sign
m
th e t e r
7
f
v
h
is
in
n i
e,
1 th e s er e s s
a n d t e re o re
s a re
p
t
os
e an
x
i
ex
u
e
a
o rt o r
so
s
e
en
at
n e
t
x
e
.
e re o re
r
c o n ve r e n
a
ro
e ri c a
n
r
e
29 3
To s he w th a t th e exp a n s io n
x i s c o n ver en t o r ever
x
v
a
e
o
l
u
g
f
y
f
f
a
Li
m
e n e
.
o
x
er
er en
s co n
a
in
t
a re
en
t
o
so
e
e o
.
a scen
din g p o wers
.
H
the
f th e
f mt h i
y
v
v wh
m
wh
v
i
g
L m
u m lly ; h f
i
m
i
l
h
s
i
f
l
t
t
f
h
t
g
h f f i i it i v g wh m f
i iv
i
v
2
t
A
t
8
d m
3
g
[
]
h
m
wy
fi t
r
.
o
a
‘
1 , t is ra t io is n e ga t i e ; t a t is ,
s a re a lt e rn a t e l
i
n t th e
t
er
o
s it i e a n d n ega t i
o
p
p
is p o s it i e , an d a l a s o f t h e s a e S ig n
e n x is
v
wh
mi l
.
r > n +
en
o
en
u
W
is
exp an s i o n
when
t
—
2
th a t the
s co n vergen
u
a n SI On
ere fo re
s h ew
To
.
Theo re
p
h
t, t
erge n
5
x
u
e re
v lu
a
u
n
e o
—l
fx;
h
10 D0
n
;
en c e
a
th e
an
d t
h f
s e rie s
e re o re
is
co n
v
u
u
er en
g
t
n
.
l
—l
wh
a t e ve r
be
HIGHER
23 8
GE B R A
AL
.
To s h ew th a t th e exp an s i o n of lo g ( 1 + x ) in a seen
294
eri c a ll
h
an 1
l
e
ss
t
ers of x is c o n vergen t wh en x is n u
w
o
y
p
m
.
H
u
is
eq
al
If
v
th e
e re
e rge n
to
n
um
s
th e
x =
1,
t
A
rt
[
e o
a
h en c e th e
x
.
v lu
e ri c al
b
f
u
s e rie s
e rie s
u
is
1
n
“
v
co n
e rge n
l —l
2
es
t w
ic
h
is le s s t
x
en
t h e li
in
1
l
i
wh h
x,
n
—l
n
m
ec o
.
l
- -
4
3
an
h
d is
mt
i
1
an
.
co n
.
m
l
1
1
— 1
—
— l
an d i s
b
c
s
r
i
s
e
o
e
h
t
e
e
e
If x
,
2
4
3
o f z er
o is
T is s ews t h at t h e lo ga rit h
dive rge n t [A rt
t h e e q at io n
in n it e an d n e ga t i e , a s is o t h e r is e e iden t ro
s
z
.
fi
h h
v
.
u
w
m
f m
u
v
.
an
u
m
E x a p le 1
Pu t
h
c” ;
x
Fin
.
t
d
mit
th e li
u
f
o
.
lo g x
wh
x
is in fi n ite
en x
lo g x
y
2
y
ell
y
—
l +y +
+
+
2
3
l
[
o
1
1
als o
wh
m
1
L et x
,
y
let y
wh
Now
h
”
h w h wh
h
h
S
.
so
t
at
z , so
t
en
is
n
at
t
e
u
.
u
en n
en ce
m
nx
m ms
fi
u mb
u
t h e p ro d
ct
e
f ux
o
”
=
wh
o,
z ero
.
1
en x
.
to
f
f
y
to
ac t o rs
c o n s is t o
f
fn
u
fi
u m
also
lo g y is fi n ite ;
.
n e c es s a r
er o
en
h
f th e frac tio n is
0
“
”
m y
al e o
z
y
if a s n in c reas e s in de n it el
ate l
b e z e ro , an d if u ”
1 t h e p ro d
a
n it e ,
o rde r t a t t h e
ro d c t
b
e
p
y
t
v u
h
u u u
h
th e
m
lo gy
I t is s o e t i
o f an in
n it e n
e
|
3
lo g z ; t en
1
lo g e
n
l lo g z
—
'
" [ lo
lo g y
z
gy
y
lo g z
in fi n ite z is in fi n ite, an d
0;
at n
Li
s
t
+
is in fin ite th e li i t
'
S ppo
1
2
y>1
erefo re
29 6
ro d c t
p
y
”f
h
is in fi n ite y is in fi n ite ;
2
l
e
p
E xa
als o
+1
y
en x
.
en
x
t
m
m s
f w
295
Th e re s lt s o f t h e t wo o llo in g e x a ple a re i po rt an t ,
d will b e re q ire d in t h e c o rs e o f t h e p re s e n t c h a pt e r
fi
m
h th
det e r in e w
is n it e o r n o t
fi
a c t o rs an
e
er
th e
.
d t o b e den o t e d b y
”
u
u
1 , t h e pro d c t will lt i
e n c e in
c t will b e in
n it e
t 1
s t t e n d t o t h e li
i
u
”
u
mu
fi
h
m
.
24 0
HI
Fo r
ex a
mpl
if w e
e,
ex
GH E R
p
an
AL
GE B R A
—x
l
)
(
d
2
"
.
b y t h e B in
o
we fin d
mi l Th
a
e o re
m
,
l
(
B u t if w e
l
a in e d in A rt
p
1
wh
+
o
.
m
b t a in t h e s u
6 0, it a pp e a rs t
2x
3x
+
2
o
h
+ nx
+
f
m
ter
n
s
o
at
"'
1
1
— x
—
x
"
)
n x
s e rie s
ex
II
l —x
2
as
’
en c e
l
— x
1
(
a rde
g
wh
)
n
a
d
u
t h e tr
as
1
(
If
x
is
n
1,
t a t i t is
x >
or
h
e e
—
we
te,
n
a le n
t
see
o
t
h
at
f t h e in
fi
th e
u
fi
an
9
—
1
(
an
e,
n
n
s
es
it e
n
n
en
on
s
es
x
<
)
re
e
s e rie s
t
n
e
ec o
te
n
e c an
a
es
n
en
x
x
-
)
<
rt
,
en
x
l,
:
so
.
a
t o in f ;
.
e
e
o
e
n
as s e r
9
h u ld b l d
i
f (1
)
f
ll v lu
2
i
3
1
i
v g
i
e x p an s o n
v
x
ly b
o n
.
an
s
to
'
e
— x
c an
2
'
x
o
x
er
e rro n e o
u
s
ui
c o n cl
2
m
if we
s on s
m
h
w
to
e re
u se
w
b y t h e B in o ial T eo re
a s if it
e re
I n o t er
x
o rds , w e c a n in t ro d c e t h e
2
in t o o u r re a s o n in g wit o t e rro r
x
en t , b u t w e c an n o t do s o
e n t h e s e ries
x
o
fo r a
a
es
in n it e s e r e s
if t h e s e r es s c o n
is di e rge n t
tr
)
l
v
i h
1
)
ti y b
m i fi it wh
i fi it
t h i qu
i i h
i d fi i ly wh
d dim
l [A
ly wh
1 th t w
t th t
1
(
s
x
2
+
1
1
d we
n x
72 93
C
B
en
fii
i
u
v
q
in
n
g
x
g
+ 2x + 3x +
l
2
mki
By
an
h
f t is
h w
.
u
hu
wh
.
u
v
hv
m
Th e diffi c l tie s o f di e rge n t s e rie s a e c o p e lle d a dis tin c t io n
a de b e t
e e n a s e ries an d it s a lgeb ra i c a l eq u iva len t
Fo r
to b e
l
e , if w e di ide l b
w
as
ex a
1
e c an
al a s o b t a in
p
y (
an
as w e
l
a s e o f t h e s e rie s
t er
e
p
m
m
m
y
htv
w
a e
a le n
ms
er x
my
a
v
b e,
.
wy
an
d
so
in
a
c e rt a in
y
wh
my
1
s en s e
hv
it s a lgeb ra ic a l eq u i va len t ; y e t , a s we
c e do es n o t re all
e x is t e x c e t
n
e
p
c alle d
v
w
a
a
e s ee n
th e
,
th e
s e ries
is
e
be
u
i
q
co n
C ON
v e rge n
as
t
VE R G E N C Y
I t is t
.
I
m
h f
o re
e re o re
f
t h e gen era tin g
DV
AND
u n c tio n
1
o
a
ER
pp
f th e
2x
3x
GE N CY
ro
p
o r SE R I E S
ria te
to
n ct o n
c
e
mg
Th e u s e o f t h e t e r
e x la in e d in t h e c a p t e r
p
h
n
R ec
u
E XA M PL E S
.
Fi
n
d wh h
et
x
x
an
d
+
+a
x
1
+ 2a
v qu
e
v qu
b e in g p o s iti
v2
x
v
x
z
6
7
e
+
ii
an
titie s
4
x
3
E
I
x
l
+
i— +
‘
Ij:
3
+
21
.
8
.
—
p
x
x
10
5
4
3
2
2
+
15
+
3
x
.
11
3
x
.
H
4
2
g
.
H
‘
l
‘
.
,
Z
g
A
.
2
s
+
l
3
f
v
-
15
+
.
l
3
x
2
3
—
x
+ 3a
tit ie s
an
1
1
1
x
.
17
31
4
e o
.
e
on
es
+
f
XXI
.
v
a
by
y
o rdin a r
.
u n c tio n
s e ries a re c o n
n
1
b e in g p o s it i
a
wi g
t h e fo llo
1
1
1'
er
e
gf
rrin
ries
S
e
g
en e ra tin
on
o
2
en
e s e r es
p
e ak
s e rie s
h fu i wh i h wh d v l p d
u s w ill giv t h i i qu ti
b e in g t a t
b ra ic a l r le
s
24 1
.
w ill b m
e
o re
.
.
e rge n
t
d iv
e rge n
t
,
24 2
HI
12
6
2
2
+
.
.
1
.
14
+
17
.
+ m +
Te s t th e
4
s erie s
wh
2
Tes t t h e
.
3
u
N/
.
1
1
x
+1
x
1
—l
l
x
e rge n
S
20
.
is
v
co n
21
S
all
hw ht
t
e
e rge n
.
t fo r
t
or
d iv
hw ht
e
t
n
ite
22
.
in
fi n i t e,
e
ex c e
t
t
.
—
2
x
v lu
a
e rge n
2p
3p
4”
3
I
7
—
4
“
es o
fp
s
n
at
+2
fi
n
t
ac c o r
ite
s e ries
di g
4
4
4
.
n
ut
2
x
e s e ries
wh i i fi i
Sh w h
wh
p wh
en n
—1
l
as
mQ/
Li
n
te
en
u
n
is
1,
1
or
.
c
2n — 2
6
2n — 2
'
2n — 3
fi
J
it
4
.
th e p ro d
a
2
is
—
1
+
+3
1
+1
x
t h e in
a
x
1
1
v
J
n
4
1
+2
x
rac tio n
e
at
e
s are
( 2)
-
v f
th
Sh w t h
.
m
ge n eral t e r
s e ries
x
b e in g a p o s iti
co n
l
-
4
o se
—
l l —n
z
1+
(2 )
is
x
n
.
+
3
2
1
( )
19
—
p
(I)
18
G EBR A
1
3
1
.
AL
2x +
.
16
3
x
fi
1
l
1p
x
14
GH E R
2n — 1
2n
°
2n — 1
.
x =
1,
n o
v
mi
t er
en n i s n e ga ti e a n
d
n
p
o
f
e r c ally
gre ate r
t
th e
um i
n
ex
an s io n
h u
an
is
n
ity
.
24 4
t
h
HI
(
is ,
at
GH ER
G
A L E BR A
v
vl
.
v
2
3
1
v
H
t
e rge n
is dive rge n t
v s e rie s
if t h e
e n c e,
-
-
di
a ls o
is
u s e rie s
th e
.
v s
h
s
h
v
W e h a e een in A rt 2 8 7 t at a s erie is c o n e rgen t
300
th
it o f t h e ra t io o f t h e n
ter
o r di e rge n t a c c o rdin g a s t h e li
is le s t a n 1 , o r gre at e r t an 1
I n the
t o t h e p rec edi n g t e r
o re c o n
e n ie n t t o u s e
a in de r o f t h e c a p t e r w e s all fi n d it
re
t i t e s t in t h e e q i alen t o r
96
.
v
m
m
ra tio
th e
le s
o r
s th
u
m
1
an
t
mil ly
Si
en
is c o n e r e n o r
th
to th e
t er
o f th e n
s e rie s
h
th e
ar
t
n c a te d
i
h
s i s w ill b
vid d h t Lim
w
h
s
t
h
t
g
Th e
u
er e
-
p
t
e
eo re
m
u
v er e n
n
-
f th e
o
m
i l
v
is di
h
.
v
mit
m
m
an
u
re c e
or
<
+ 1
din g
v
-
s
a rt ic le
e rie s
is
d th e w s e rie s
t p ro
e rge n
n
h
1,
>
th e
+1
Li
.
ac co r
en
v
”n
--
p
t w
e r en
g
Li
>
.
e rie s
e v
en
v
e co n
a
su
m
din g a s th e li
is gre a t e r t
cceedin g t e r
t
e rge n
acc o r
u
ro
m
din g a s Li
is ,
at
m
s h
h
f m
div
t
h
uv
v g
hs
A
.
v id d t h
e
1
f
1,
o
an
.
my
a
co n
v
w ill
be
e rge n
t
b e di
at
m
7)
”
U
*
3 01
vergen
t
L et
wh
o se
W
c a se
Th e
.
co
us
mp
m
en
p
>
en
v
s
u
h
at
u
1
v
y
i
x il a r
co n
v
h
g
is
1
1
n
is , if
is , if
wit
t if
"
n
at
s
s e rie s
e r en
1
t
s e rie
en
2n
>p +
2
s
>
p
“n +
t
“
is
1 th e a
is
e rie
v s
t h e gi
m
mi
ter
1
t h e gi
a re
t er
r
n
e
e
al
g
Li
a c co rdin g a s
n e ra l
e
g
h
s who s e
s erie
+
un
is
1,
o r
h
th e
co n
v
c o n vergen t o r
<
1
a
e rge n
di
.
u
ys
x ilia r
t,
an
e rie s
h
d in t is
VE R GE N C Y
C ON
y
u tity h
b lis h
Bu t th e a
b y a n it e q
ro
o s it io n is
p p
in g a s
H
e s ta
is
ER
co n
v
v
ergen
ere
Li
t
or
mi
;
v
e s eco n
m
u“
th e
er
h
n
“n
th e
O
en c e i f x < 1
h
In t is
.
vergen
t,
Le t
m
u s co
1)
2
n
(
+1
h
on
.
0
seri e s i s c o n
v
erge n
t,
an
0
if x >
d
l
n
'
a
n
n
as
fi
( n
1)
(2 n
2
+1
is
s erie s
a re
1
u“
co n
v
ergen
t
.
mi u
wh o se gen era l ter
Li
m
t h e gi
1
it
c ase
mp
t
o
+ 1)
( 2n
2n
u
m
g
ro
—2
—1
2
2
n
n
)
(
s eries
acc o rdi n
is
W
The
0
—3
2
n
(
)
“n
.
e
c as e
1
302
t
o
.
=1
1 th e
ar
ro ce e d
an
,
s erie s
u
en ce
e r en
s
.
“n + 1
Li
h
.
e
2;
t
a
ro
ergen t
+1
ergen
er
a
div
n
we
ed
o
ser e
1
If x : 1 , L i
ter
24 5
.
t if p is gre at e r t an 1
en c e t h e
r t
a rt o f t h e
p
ar
et
u
is di
s eries
*
o r SER IE S
e rge n
x
n
.
i
h
GEN C Y
e a
e
e o re
[
th e
an
en
E x a p le
co n
s e rie s
x ilia r
I
v s mll ; h
fis
li
i
h
i
i
i
1
t
u
y
s
d
v
t
d by p
p
g
w my p v t h
d p t f h p p si i
b f
m Fi d wh h
fi
Wh
is
u
DV
AND
lo g
n
v
u
1,
ii
s e rie
en
s
s w it h
is
,
1
or
the
co n vergen
t
,
s e ries
wh
o se
n e ra l
e
g
p
en
t h e gi
p
v
>
en
l th e a
s e rie s is
u
i
ys
x il a r
co n
v
e rie s
er en
g
is
t if
co n
v
er en
g
t,
an
h
d in t is
rt 300
A
]
[
.
t
h
at
is , if
1
l
o
p
g
l
2 46
t
h
HI
H
h
a
u
xa
lo g
fi
en
y
th e
a rt o
rs t
v
m
pl
erge n
t
et
dv
i
or
th e
er
hs
c ase
s eries
erge n t
5
4
[
n
.
.
d wh h
Fin
e.
v
co n
.
sm m
3
is
GE BR A
h
f
t
e p ro p o s it io n is e s t ab lis h e d
p
in t i
an n e r
i
il
ar
w
e p ro c e e d in a
1
p
s e rie s is di e rge n t
x ilia r
E
m
I/t
en c e
W
AL
'
is , if
at
GH E R
I
I
.
n
x
n
[
it
[A rt
H
if x <
en ce
é
th e
series
is
v
ergen
co n
t, if
x
>
1 th e s eries
is
e
1
lo g e — n lo g
—
z
2 20
.
div
ergen
t
.
)
— 1 —n
Li
h
en ce
*
t e st
wh
th e
en x
3 03
s giv
m
“
th e
en
p
s wh
v
co n
ro ve
s 3 00
fu th
n
in A rt
v
er a
o se
d in
g
th e
y
.
or
erge n
d
an
t
“n + 1
.
m
a ls o
Li
+ 1
3 01
,
r
n e ra l
e
g
e r en c
(
lo g
n
div
is
— 1
,
I f Li
.
To dis c o
s e rie
series
m
N"
di
er
t e s t we
mi
ter
v
a re n o
n ex t a rt ic le
.
a
l
i
a b le
c
pp
.
h ll mk
a
a
1
s
e rge n c
s
t
y
l
o
n
( g
of t i
n
)
”
hss
e u se o
In
e rie s
f th e
o rde r
we
n e ed
to
a
u
e s ta
th e t
y
b li h
x ilia r
h
s
e o re
m
24 8
HI
*
.
if
1,
p
series
Th e
3 05
an
GH E R
AL
G E BR A
mi
wh o s e gen era l ter
d divergen t if p
1,
=
or
<
p
.
1
s
n
1
l
o
Ii p
( g )
sw
mv u
m
Th e
c o n s tan
f
t
a c to r
f h giv
w
i
v lu f p th
i
l
ll
u
u
f
w
d
t
q
t
o re
e
o
es
a
re
en
o
Th e
.
t
a cco r
Le t
u s co
ver en
g
m
t er
o
s
s eries
din g
wh
mp
co
mm
e r en
g
on
t
or
e n e ra l
g
co n
s wh
to
di
e
v
e rge n
t
1
is
t
H
?
or
g
o se
e n e ra l
g
t fo r t h e
e re
sa
m
e
en c e
th e
t
di
.
s u
n
is
c o n vergen
m
1
e r en
er
er
n
mi
v
v
v y t m; h
m
ter
wh o s e gen era l ter
t h e gi
a re
v
o se
A
rt
[
.
Li
as
is
?
e s e ri es
re s
3 06
*
l
o
a
( g )
ill b e c o n
s e r es
as
re
1
t
co n vergen
.
ill b e
B y t h e p re c e din g a rt ic le t h e s e rie
di e rge n t fo r t h e s a e al e s o f p a s t h e s e rie
te r
is
v
is
1,
>
en
s e rie s
i
y
wit
h
th e
s e rie s
wh
o se
or
o r
1
.
e n e ra l
g
is
n
W
c ase
h
en
t h e gi
p
v
>
en
th e
1
s e rie s
a
is
u
co n
(
N
o
w
wh
en
lo g ( n
H
t
h
en ce
at
is ,
t ha t is ,
th e
co n
n
1)
is
=
x il a r
lo g n
dit io n
s
is c o n ve rge n t ,
A
rt 2 9 9 , if
e rge n t b
y
an
.
1 ) {lo g ( n
”
n
l
o
n
( g )
n
v yl
er
v
s e rie
a r e,
g
lo g
1
b
ec o
( )
1
+
m
es
1
n e a rly
;
h
d in t is
VE R G E N C Y
C ON
DV
AND
I
ER
GE N C Y
b gn
s ec o n
fi
th e
en c e
d p a rt
m
E x a p le
.
a
I s th e
>
p
.
h
h
ro o s it io n is e s t a b lis e d
f
t
e
T
h
e
p
p
p
a n n e r in dic a t e d in A rt 3 01
ro ve d in t h e
p
art
rs t
my b
24 9
.
1
l
H
o r S E R IE S
e
o
m
.
.
.
s eries
22
co n
v
e rge n
t
or
dv
i
ergen
t?
2
( n+
un
“n +1
1,
F
ro
an
d we p ro cee
—1
m
an
=
Li
{
m
0 [A rt
1
h
.
hv h
mh m
v wh
fi
u
u
.
1+
d we p as s to th e
n
)
n ex t
te s t
.
lo g n
1
en ce
'
tes t
n ex t
1
s in c e
4n
n
to th e
m
1,
Fro
d
1
1
5
2
( 10
1
)
1
lo g
v
th e gi
n
en s erie s
J
dv
is
i
e rge n
h
m
v
t
.
v
W e a e s e wn in A rt 1 8 3 t at t h e u s e o f di e rge n t
307
at e
l
s e rie s in
a t ic a l re aso n in g
a
e a d t o e rro n e o
e s lt s
s
r
y
n it e s e rie s a re c o n
B u t e en
e n t h e in
e r e n t it is n e c e s s a r t o
g
s in
t
h
e x e rc is e c a t io n in
e
g
96
.
.
Fo r in s ta n c e , t h e
l —x
is
co n
s e rie s
v
er en
t w
g
b y it se l
f
,
h
en
th e
m
u
u
y
.
.
s e rie s
x
x
“
3
+
x
x
5
5
3
:
co e
ffi
A
rt
B
u t if w e
[
“
c ie n t o f x
in t h e p ro d c t is
1
.
.
u
mu lt iply
the
2 50
HI
hs
D en o te t i b y
a
1
/
d is t h e re
an
a
If
a
—
o
a
l
T is le a ds
n it e
o f t w o in
308
*
co n
us
L et
.
a
an
gn ,
0
n
u
i
q
v g ts
h
en
e r en
a
m
“
wh
de r
is a ls o
n
e rie
2
s
+ a3x
,
b ,x
in
a re
de n o t e t h e two in
+
is in
n
fi
n
it e
.
es
u
re
n
+
2
+
e
fi
+2
n
sh
s e rie
it e , t h e
n
as
bs x
3
3
fi
n
+
at
co n
co n
it e
v
e r en
g
t
s th
e
p
ut
d
c
ult
o
ro
.
s e rie s
+ a
b
+
dit io n
x
x
2”
2"
+
.
e e
f m
it e w
.
en
ec
re
n
m
, "+ 1 ,
sp tiv ly
multiply t h s s
dB
I f we
th e
a
+ a x + a2x
60
by A
a
to
us
fi
fi
in
o re
+ a2
ms
m mg
hm
(U )
7?
b ec o
c
o r
+ ag
d s in c e t h e t e r
e t ic a l
ea
n o a rit
h
f
ut
l , t h e pro d
x =
an
>
—r
2
n
f
f/
.
n ce
en
1
r
GE B R A
AL
h si
t
,"
GH ER
t o ge t h e r we
e rie s
b t ain
o
a
re s
f
or
a
(
b
o 0
a
b
, o
a
b
c ,)
(
x
a
b
g 0
“
9
1 1
a
b
c ,)
x
2
u s hs s s
fi
m u
h
hv
wh
m
u
hm
uv
u
F i t su pp s t h t ll t h t m i A d B
i
t
i
v
p
L t A
B
t th
i
f md b y t ki g th fi t
0 d
2
1 t m
s f A B 0 p ti ly
If w m
u ltiply t g t h th tw i A B th fii i t
f
h p w f i t h'i p du t i qu l t h ffi i t f
h lik p w
f
i C
f
th
m
b t i A E
th
t m
t i i g p w
h igh t h
wh il t
f
i t h h igh t p w
h
f i 0
e rie
t o b e c o n tin u ed to in n ity an d le t u s
S ppo e t i
at c o n dit io n s
de n o t e it b y C t e n w e a e t o e x a in e n de r
a
t
i
c al e
i
a le n t o f t h e
b
e re a rde d a s t h e t r e a rit
e
C
y
g
q
ro d c t A B
p
.
rs
o
e
e
a
en o
2 ",
er
n
o
,
e ac
t
e
o
o
e
o
e re a re
x
2”
s
er o
er o
er
e
x
x
s
co n
es
o
e
n
e s er e s
ve
ec
or
o
ser es
e r
ro
c
s e
as
n
ar as
o
er o
x
u
e rs
n
a re
e
a
h
m
.
rs
e
n
2"
e
o
a
t er
x
e co e
2 ",
t
o
e
x
er
co e
an
n
x
c en
c en
u
o
M
“
2”
s
,
en c e
11
m b
m th
I f we o r
t h e p ro d c t A a t h e la s t t e r
is
0 2 " in c l de s a ll t h e t e r s in t h e p ro d c t an d s o e
en c e
b e s ide s
C >A B
u
e
os
.
e
A B2
f m
an
n
er
n
a n
s
er
e
res
,
e
o
e
a
u
a
73
o
‘
x
”
l
er
t
u
b
;
m
te r
s
2 52
HI
*
Fin
d wh h
et
1
1
x
AL
E XA M PLE S
f ll wi
th e
er
GH E R
o
o
n
2
g s e ries
G E BR A
XXI b
.
.
are c o n
v
.
x4
div
t
or
wh
ere
e r en
g
e rge n
t
s
a
°
3
2
2
3
”
H
4
.
'
l+
2 2
3 x
2x
—
2
4
5
.
2
i
7
.
x
4
+
I5
B
3
L
2
'
4
5x
x
li_
l__
1
—
3 3
l
is
3
.
2
1 +
1
(
a
— a
1
2
1
2
2
)
—a
2
(
)
2
1
a
.
b e in g a p ro p er frac tio n
a
(
+x
°
-
.
2
(
a +
3x )
3
d
(
a
+ l l is (13 4 1 )
-
1
10
.
x
o
l
( g
an
12
°
u
3
I
1+
2
2
2
2
1
9
2x)
a i
-
2
x
3
k
"
An
n
m1
hw h
v
4
x
l
o
( g
‘
1
Bn
2
.
3
.
l
o
( g
k
‘
z
a
'
3
.
t
in teger, s e
A
a
1 is p o s it i
at
th e
e , an
d
s eries
d iv
erge n
u1
u
t if A
u3
2
a
1
is
is
n e at i
g
v
la is
co n
a
v
p o s iti
ve rgen t
e o r z e ro .
e
if
C H A PTE R
UN
D
E TE RM IN E
D
XX II
.
C OE FFI C I E N TS
.
m
v
3 09
I n A rt 23 0 o f t h e E le en ta ry A lgeb ra , it wa s p ro e d
t a t if a n y ra t io n al in t e gra l
n c t io n o f x van is e s
en x = a ,
it is di i s ib le b y x
S
e e a ls o A rt 51 4
a
C
on
[
]
.
h
.
v
6
i
b
v i h s wh
an
Y
’
p ox
rat o n a l
a
s
.
e
en
+
in t egral ; f
n t
n
of x
p e’ io
x is e u al t o ea eh no f t h e
q
‘
fu
hv
D en o t e th e
we a
by x
a
u
a2
i
n ct o n
by
o
f( )
ar
,
o
nc e f (x )
si
1 di
e n s io n s
o
tien t b e in g
p ox
si
u
an
c
t itie s
x
a
f( )
n ce
x
v
is di is ib le
o
m
fn
2 di
(
2
“
h
s
f( )
=
If a
ra
x
3 10
m
.
o re
th a n
va ria b le
L e t th e
fu
n
p0
(
CC
— Cl
tio n a l
va lu es
mtb
us
i
n ct o n
.
v
en
x
) (p
“
on s
an
a,
si
a
Pro c e e din g in t is wa y , we
f
of th e
en
is di is ib le b y
(
or
h
m
fn
x
vis io n
t
u
,
wh i h
e n s io n s ,
.
n
x
m
e
tie n t b e in g
mil ly
u
u
3 ,
fn
fl
di
n e
al
q
q
o
“
,
p
(p
Si
th e q
.
"
a
th e q
h wh
.
/
Le t
e
fu
3
s
) (p
h
(
)
a
IE
o
o
ll
— (l
we
h
a ve
2
-
f
fi
z ,
d
3
y
n a ll
)
o
b t a in
(
a
a
f
te r
n
di
CU
m
in tegra l f un c tio n of n d i en s io n s va n is h es
of th e va ria b le, th e c o e ic ien t o
f ea ch p o wer
fi
e z ero
.
b e de n o t e d b y
f( )
+p x
x
wh
e re
HI
GH E R
f( )
v
is
a
a
2 54
an
u
su pp
u
v
l
q
d
n e
t h at
o se
a
a
ues
l
a ,
f( )
po
b
t
x
L et
s in c e
c
f( )
c
e an o
f
h f
s i qu
d t
(
s e
v
u
pl
l
h
p
2 ,
s
”
h
T is
re s
2
)(
—
c
a
)
,
o
f th e
(
a
e
(
—a
)
a
x
)
.
h ; th
va n is
en
c
)
0
=
o
f th e
o
th e r
u
v
an
is
h s f m th
e
or
o re
an
v
n
ues
al
.
my h w t h
we
e e
u
q
a
s
al
to
z e ro
en
u
be
e
a t e ac
h
f th e
o
fi
c o e f c ie n
t
s
.
n c ia t e
d
u
i
n
t
e
r
a
l
n
c
t
i
o
n
o
f
f
If
g
o re th an n va lu es Qf th e va ria b le, i t
Qf the va riab le
m
h
e ac
y h s
ra tio n a l
a
)
c
0
a so
a
s
x
w h i h mk s f (
e x p re s s io n
an n er
ar
2
x
to
+p x
mil m
m
u
t
b
p
u lt my l
i
f
al
en
2
y h
h f
a s
o
u
q
e
b y h po t es i , n o n e
H en c e f (x ) re d c e s t o
.
B y h p o t e s is t is
o f x , an d t e re o re p 1
In
e
h
is
x
a
s in c e ,
z e ro
a;
t
.
e
0,
-
wh e n
es
a
—a
l to
h
3
e r va l
p0
a
an
—
0
e re o re
a cto r
h
(
2,
x
0, we h a
z
Po
an
x
G E BR A
AL
as
f
llo ws
o
m
i
u s t va n i sh
11
m
di
va n is h es
en s o n s
f
f
or
va lu e
o r ever
y
.
I f t h e f un c tio n f (x )
equ a tio n f x
h
as
0
( )
C OR
o f x, th e
.
H
ro o ts
it is
an
m
E xa p le
iden ti ty
Pro
.
i
x
(a
h
u
v u
T is e q
o f th e t h ree
a
n
f
i
a ls o ,
en c e
v h
atio n
f
or
u s
S pp o
u
q
a
o re
f
11
o
es
m
fo r
th an
o re
n
ro o
m
di
en s io n s
t
s
t
h
.
h as
c
)
dim
h
x
c
—a
-
a
)
i
(c
x
i
a
a
en s io n s , an
en c e
c
c
an
n
m
o re
v
a
u
l
e
s
th an
n
it is
d it is
i en tity
by
h
d
an
< b)
>
x
>(
—b
e
en
c
vid
)
tly
d
satisfi e
eac
.
m
e
ra tio n a l
.
fu
t h at t h e t wo
po
a re e
h
in tegra l fu n c tio n s of 11 di en s io n s a re
o re th an n va lu es of th e va ria b le, th ey wre e u a l fo r
q
o f th e va ria ble
m
e very va lu e
-
Of two
b,
m
>
<
—
b
b
< ><
x
al e s a ,
is
at
a
is
an
.
<
>
—
—b
(
>
—b
If two
equ a l
t
e
equ a ti o n
v
l fo r
m
x
qo
o re
"
x
+
"
t
h
p
x
n
n c t io n s
—l
+
2
"'
2
“
p 2x
x
q,
an
"
q2
u
x
v al e s o
n
x
n
+
p
q
f x ; t h en
—2
th e ex p res
+
+
si
(p
on
_
Q
)
G
2 56
AL
H I H ER
H
en ce
th e
su
m
2 71
A
To fi n d A , p u t
1 ; th e
,
3
s eries
t
5
re
en
2,
A
2
h
.
1
2
n
-
GE BR A
du
an
,
d
0
A
or
m
t o its fi rs t ter
ce s
H en c e
w
u
wh h
F d
mthi
it i
dim
N OTE
I t ill b e s een fro
ra tio n al in tegral f n c tio n
of n
f n c tio n o f n
ic
is o f o n e
.
u
s erl e s
.
m
E x a p le 2
in
.
atin
mth
c o e ffi c ie n
g th e
ts
at
h
la s t
e
eq
u
atio n
1
b
+px
2
th e n
is a
ter
e fo r th e s u
a
ter
o f th e
th e ri
en
as s
an
qx + r
my b
a
e
dv
i isib le b y
.
th e
co n
di
tio n s
+
h
p
—a
(p
,
)
u d
req
ire
a e
.
5
a_
hv
p o ers o f as, we
a k + b = q, kb z r
e
,
by
en c e
an
d
an
d
ar
b
s
u
u
b s tit tio n we
et
ar =
b (q — b ) ;
XXII
.
e er
n
o
o
.
a
oe
ne
.
ts t h e
c en
su
mf
o
m
t er
n
.
b tain
+b_ q
d b y th m h d f U d t mi d C ffi i
e
o
.
E XA M PLE S
n
at
k w
7
k
b
r=
a re
Fi
3
x
at
ax
f th e li
’
is ,
ic
2
s
t
ons
k+ a =p ,
b
h
wh h
t
o
r
t
t
e
Eq
Fro
di i
co n
s
m
m
m m
h wh m
u
u
h h h
e
ex a
ffi c ien t to
ig er t
e n s io n
,
x
um
u
A ss
th e
mpl
s
th
s
.
0
5
9
5
2 4 2
5 +
n
n
t er
s
.
9
n
ter
s
.
7
8
.
O
.
.
.
t
n
perfec t
9
d h di i h
f h f m +
Fi d h
di i t h t
Fi d h
di i h
qu
Fi
t
e
or
t
n
are
P
2
2 ax
t
a
ons
t
co n
at x 3
t
on
ons
s
.
2
3p x + 2 q
my
a
be
divi ib l
s
e
by
a
.
a
t
ax 3
bx
2 4
a x
at
2
car
d
my b
a
2
+ b x + cx + dx
3
e a
t
c
r
f
ec
e
p
f
2
my
a
ub
be
e
.
a
.
t h at
ve
p
p=
2=
ac , a
x
e
t
co n
e co n
t
n
s
e
m
te r
m
m
2
fac to r
1f 6
2
-
5
6
‘
2
1
a
j
2
+ 2 b xy + cy + 2 dx + 2 ey
fi
é =
a , e
c
‘
z
f
is
a
p erfe c t
sq
u
are,
C O E FFI C I E N TS
U N D E TE R M I N E D
d ivi ib l
I f aa + b x + cx + d is
2
3
10
.
11
.
If
12
.
Pro
x
v
d ivi ib l
5qx + 4 r is
5
d
th e i
e
e
2
—c
(
x
)(
c) ( b
d
Fi
n
.
th e
co n
th e p ro
e
a
du
—a
x
a
t
2
c t Of t
o
sa
m
c
an
If
u i
h g d wi h
u
t
e
x,
3
y,
l +2
2
15 S
n titie s
a
q
u
.
hw h
e
a, a
t
at
2
3
(
a
,
a
(
313
to
z ero
th en
th e
fi
y
f
o r e ver
all
a
e
s
,
fi
n
i
i
en t
c
fi
ea c h c o e
(
)(
a (c
)
—
x
b)
(
a
n
a
r
f
zy H
— c
)
o
0,
r
o
p
e
+2
1) m
(
a
x,
n
du
1)
n
l)
s eri es a
va lu e
m
u st
qf
x
a
f
b e equ a l to
2
%(
-
—
n g—
l lz , an
-
z wh
d if t h e
r , 5a re in t e r
,
5)
en
m0
+ 2l
n
1t
a x
,
or
x
y,
c ts
a x
o
ml
z
at
t
2 ln =
.
{
o
2
i te
i te
x — a
’
es
e
m+
m f th
2
l,
su
1)
1)
r
m
is
a
(
x — a
(c
)
v lu f
v ly h w h
n =
If th e i n
.
2
0
)
f
e
m
l
4
by + 2gx 2fy + c
ac to rs o f th e fo r
wo
,
r t-
.
2
+ 2 hxy
fo r
re s p e c t i
tr
a t o n s a re
e eq
be
at a d
5
at
t
e
t
at
p H
.
,
e
.
’
14
s
,
ro
)
(
d) ( b — a )
—
—
—
—
—
a
x
x
b
x
a
v
b
v
d
x
)
(
)
(
)
)
(
)(
(
(
d i i n th
ax
my b
2
)
h
p v
hw h q
h
d)
.
13
c
x
) (x
—c
)
(
)(
at — b ) ( a
(
x — c
.
(
by
s
x
2
titie s
en
b
v—b
by
e
s
2
2 57
.
=
.
to ge t
r
h
er o
f th e
n
—
n
r + 1l
(
)
—r
2
.
a x
2
3
is
3
wh i ch th e s eri es is
z ero iden ti ca lly
equ a l
gen t,
c o n ver
.
L e t t h e s e rie s b e de n o t e d b y S , a n d le t S l s t a n d fo r t h e e x
2
a x
t
en
re s s io n
a
d x
a
a n d t e re o re ,
x
S
,
p
3
0
2
l
f
o r a ll f
i
n it e
xS,
by
O
al e s o f x
o t e s is , a
B
u
t
i
n ce
s
p
0
n it e li
is c o n e rge n t , S , c a n n o t e x c e e d s o e
it ; t e re o re b y
a
b
t a kin g x s all e n o g
xS
e
a de a s s
a ll a s we
l
a
se
e
y
p
l
is “0 3 b u t
is a lwa y s z e ro , t e re o re
I n t is c a s e t h e li it o f
a
st b e e
al t o z e ro ide n t ic a ll
q
o
h S
h f
v u
S
mfi
m h f
m
m
S
h f
hy h
v
m
uh m
m S
h
u
y
mu
m w h v S O f ll fi it v lu f
R mv i g t h
i
f
h
t
i
v
h
s
ll
fi
i
v
u
t
t
l
f
;
my p v i su
h f h
il ly w
i
th t
Sim
l
ffi i
i
u
l
y
i
d
t
i
l
q
.
.
.
e
x
o
s, a
a
e
n
l
ar
co e
c en
H
.
ts
H
.
2
A
.
o ,
e
a
a
e x
a
z
an
d
e
a
a
a x
d x
,
a
ter
ro
e
s e
s
e
or a
cc ess o n
n
a
,
o r a
to
z e ro
en
n
n
a
ca
e
e
a
eac
.
a
es o
es O
o
x
t
.
e
2 58
HI
If two in fin ite
GH E R
AL
s eries
GEBR A
.
l to o n e a n o th er fo r every
r wh ich b o th s eri es a re c o n ver en t, th e
n
i
t
v
l
u
o
t
a
r
i
a
b
l
o
i
h
e
a
e
e
v
e
f
f
f
g
ici en ts of like p o wers of th e va ria ble i n th e two s eries a re equ a l
c o efl
314
.
a re equ a
.
u s th
S pp o
e
at
th e tw o
a
an
d
h
en
t
v
A ,x
v us
Ao
-
O
t h a t is ,
ro
o,
z
a
wh i h p v
=
O
A
x
.
n
,
a1
,
( 12
w it h i
Al
=
0,
3
o
an
n
x
a
Q
—A
z
n e
g
O,
=
2
A
z
3
a
z
a
3
)
x
3
+
mit
d li
—A
s
; t
h f
e re o re
O,
=
3
A
.
d
a s erie s o
In
d
f as c en in g p o
w
e rs o
fx
as
far
5
.
x
—
v—x
a0 + a
1
x
+
a
2
x
+
a x3
3
+
.
are c o n s tan
,
a
a
—A
S
a s si
th e
n
l
a
l+
ere a o
A gx
x
s
E p
m
i v lvi g
t m
er
—
al
2+
wh
2
f
o
e
t h e p ro p o it io n
es
E x a p le 1
th e
A 2x
3
(
h
c
z
3
o
is e s fo r all al
b y t h e la s t a rtic le
a
d x
a x
,
0
de n o t e d b y
e x p re s s io n
th e
an
as
d x
AO
a
s e ries a re
ts
wh v u
al es are
o se
to b e
d md h
e ter
in e ; t
en
— x2
)
h
h d
h
m
u
u
hh d d
h h
k
w
fi c ien ts o f li e p o ers o f x o n
ay e q ate th e c o e f
I n t is e q atio n we
“
i
h
f
fi
i
n
i
n
e
t
e
c
o
e
c
e
t
o
f
x
is a n + an _ 1 — an _2 ,
s
On th e r g t a
e ac
si e
2
ig es t p o er o f x o n th e left , fo r all al es o f
a n d t ere fo re , s in ce x is th e
a e
n > 2 we
-
.
hv
an
h w u
t is
b e en
+
u
w
v u
0;
—1
an
v
ill s ffi c e t o fi n d th e s c ces s i e c o e ffi cien ts after th e
Ob tain e
To e ter in e t es e we a e th e e q atio n s
d
.
m
d
a0 =
a
2,
3+
a
+
a
a4
a
2
3
a 5+ a 4
1
h
+
—a
—a
l
z
a
0
=
z
o,
z
o,
—a = o
,
3
hv
0,
a
wh
wh
wh
2
—
2
2 x + 5x
2
u
+
a
l
—a = l
;
o
en c e a s
en c e a 4 =
en c e a
5
7x
3
12 ;
19 ;
4 —
5
2
1
x
l
9x +
+
h
t
ree
26 0
HI
GH ER
AL
E XA M PLES
E x p an
as x
3
d
t h e fo llo
wi g
ex
n
G E BR A
.
XXII b
.
.
re s s io n s
p
in
as c e n
.
di g p w
n
f
o
as
x
6
a
7
Fi
n
.
my b
e
3
n
(
Fi
n
.
8
d
d b
a an
"
2) x
d
b,
a,
t
so
h
at
t
c so
y be
a
n
2
+1
h
s
3
If cx + a x
—
hw h
t at
3
e q at io n
x + 1 00x
res lt co rre c t 2
u
u
10
umb
s
e
y
0,
=
c o ef c ien
hw h
s
t
e
t r+ %
en c e
mi
ter
n
th e
o
"
x
ex
n s io n O f
a
p
f
—
2
x
hw h
t
e
y
c
a
15
y
1
%
T g
-
x
4
in t h e
e xp an s io n
y
'
F
(fl
.
e o
e o
3
1 2c
y
f
f y is
f x is
7
“
)
a
a
an
mt
r
x
i
o
pp
my
a e
p lac es
an
o
ui
so l
t
ml
f dec i
on
a s
f th e
is t h e
o
I n th e e xp an s io n o f ( l + x ) ( l + a x ) ( l + a 2x ) ( l + a 3x )
th e
1, s e
e r o f fac t o rs b e in g in fi n it e , an d a
t at t h e c o e ffi c ie n t o f
hw h
.
1
11
.
W
h
(
en
1 —a
) (
1 —a
z
)(
1 , fin d th e
a
l—a
3
)
(
c o effi c ie n
r
—
1
a
’
)
§
a
r
t
o
f x " in t h e
3
)
t o in f
—
wn
e x p an s io n o
f
l
1
(
12
.
I f n is
l
( )
(2)
s erie s
(3)
4
( )
th e
o
+
a
5
0 09 9 9 9 9 9 is
1= O
TO h o w
a
v lu
at o n e
z
3e
3
v lu
at o n e
x =
—
t
.
.
.
n
fi
th e
at
«
9
th e
th
°
1
‘
3
—
(1 x)
th e
far
.
2
3
—
—
l + ax
ax
x
n
e rs
.
1
H
o
s e rie s
u
n
+1
n
n
—
in
1
(
— n
eac
v
) (1
p o s it i e
a
n
ax
-
h
(
2
a x
) (1
in t eger,
s
d x
hw h
(
u
l l)
-
(
n
cas e
b ein g e x ten
“
z
(
n
)3
dd
at
—
n
—2
E
—1
—n
t
e
.
to
)
n +
n
m
n
t er
b ein g e x t e n
dd
e
%
n
s
an
d
n
+p
in th e las t t wo
c as e s
e
to
n
m
+ 1 te r
s
.
1;
C H A PTE R XXI I I
.
PA R TI A L FR A C TI O N S
m
y
.
f
u
3 15
I n e le e n ta r A lge b ra , a gro p o f ra c tio n s c o n n e c t e d
b y t h e s ign s o f a dditio n a n d s b t ra c t io n is re d c e d t o a
o re
i
l
l
d
n t o o n e s in
l
si
o
r
b
i
n
l
e
c
t
e
ra c t io n
o
e
e
b
e
c
o se
p
y
g
g
de n o in a t o r o f t h e gi e n
on
de n o in a t o r is t h e lo e t c o
ra c t io n s
B u t t h e c o n e rs e p ro c es s o f s e p a ra t in g a rac t io n in t o
r
n
i
ro
a
f
si
ac
t
i
o
s
s
O
t
i
o
l
r
o
r
a
r
l
e n re
re d
r
e
t
i
a
F
o
,
,
g
p
p
p
q
3 5x
l
I f we \ v1 s h t o e x a n d
I n a s e ri e
f
ex a
e
O
as c e n d
,
p
p
4x + 3x
1
in g p o we rs o f x , w e
ig t u s e t h e
1 , an d
e t o d o f A rt 3 1 4 , E x
s o O b t a in a s
an
t e r s as we p le as e
is t o fin d t h e
B u t if w e
li
i
n a
c a b le , a n d it is
e n e ra l te r
o f t h e s e rie s t is
i
t
o
d
s
e
pp
g
h
e
i
a le n t
e
si
l
e r to
x re s
t
h
i
i
n
t
or
e
e
n
a c t io n
e
r
p
q
p
g
1
2
"
"
an d
l
3
E a c O f t h e e x p re s s io n s ( l x )
x
(
)
—
l
x
1
3x
ia l T e o re , a n d t h e ge n e ra l
c an n o w b e e x p a n de d b y t h e B in o
.
f
m f m
m
u
f
m
m
f
f
f
mh
my m
m
s
mh
h mh
f
h
mbt
o
i
a n e
.
.
v
m
u
.
.
w h
uv
f m
m
h
d
wh
v
s
2
m
ter
u
m
mm
ws
v
m
.
u
.
h
h
m
m
v
f
uj
I n t h e p re s e n t c a p t e r we s a ll gi e s o e e x a p les
ill s t rat in g t h e de c o p o s it io n o f a ra t io n a l ra c t io n in t o p a rt ial
lle r dis c s s io n o f t h e s b e c t t h e rea de r is
ra c t io n s
Fo r a
’
’
re e rre d t o Se rre t s C o u rs d A l eb re S u é rieu re, o r t o t re a t is e s o n
g
p
t h e I n t e gra l C a lc l s
I n t es e
o rk s it is p ro e d t a t a n y
a
re s o l e d in t o a s e rie s o f p a rt ia l rac t io n s ;
e
ra t io n a l ra c t io n
b
y
i
n a t o r t e re c o r
a n d t a t t o an
li
n ea r a c to r x
n t h e den o
i
a
y
A
an
l
i n ea r
n d
re
a
a rt l al
ra c t i o n
r
t
o
o
h
o
f
o
e
t
y
p
p
u
f
f
3 16
m
.
s
s
two
i
a
r
t
a
l
p
m
es,
t
h
u
h
v
.
f
w
e re
o cc
f
u
rrin
ra c t io n
is
an
a
Bl
,
x
b
an
dditio n al
t h e de n o
d
f
h
f
h
m
f m
t
i
ce in
w
g
s
v
-
f
b
ac to r x
ti
m
f
h
f
fu
uu
.
mi
(
b)
ra c t io n
a
n ato r
B2
x
x
2
If
x
t
h
e re c o rre s p o n
6
an
d
o cc
so
u
s
tIwee
r
on
d
.
To
26 2
an
f
u
q
y
a
ra c t io n
f
dra
tic
t wic e , t
H
h
is
e re
or
2
f ll w
o
s
h
ll
a
f
q
f
a ct o r x
P1 x
ra c t i o n
(
x
2
p
2
on
ds
x
+
p
+
an
q)
x
p
p
a
a rtia
'
Q1
P, Q , P
B
B
B
c o rre s
e re
o cc
q
d
Q
so
on
a re
l
u
l
rs
.
a ll
mk
a
u se
e
o
f t
h
re s
ese
u lt
in
s
th e
ex a
mpl
es
t
h
at
.
Sin ce th e
Sep arate
.
in to p artial frac tio n s
6
d m
in ato r 2 x
en o
11
5x
2
(
6
x
11
5x
2x + x — 6
A an d B
in e
m
d
are
q
u
an
tities in
d
d
ep e n
.
as s
um
e
B
+
+2
x
we
2 ) (2 x
x
A
2
e te r
h
.
.
m
e re
t
if th e
px
u
E xa p le 1
wh
d
GE B R A
q
Q
+
t h e q a n t it ie s A
de n t o f x
We
AL
+ px +
d p a rt i a l
a seco n
e re
in de p e n
x
2
Px
f m
f th e
o
a cto r
x
o
GH E R
H I
en
2x — 3
t
o
f
’
wh v u h v
al
o se
x
es
a
e
to b e
.
Clearin g o f fractio n s ,
5x
h
w
wh
po
u
hu
Sin c e t is e q
ers o f x ; t
d
is i
atio n
s
1 1 = A ( 2x
u
tic ally tr
en
2x + x — 6
We
is
er
i
b
?
+
)2 )
(
m
—
x
e
(
an
1
( ) pu t
n o
w eq
dB
x
a
d
u
z
ate c o e ffi c ien
in
o,
d
w
ts
b
0,
k
f li
e
.
B
in g
an n er.
d
en
=
a
; t en
t
o
f x , we
,
h
m+
a
a
my giv
a
e
)
al
es
to
x an
n
+b
mb —
orx
v u
d fi n d th e
m
an
ep en
or x
o
x
a
u
2x — 3
A
x
in th e fo llo
are
ts
'
in to p artial fractio n s
n
A
p ttin g x
ate c o e f
fi c ie n
—a
to p ro c ee
Sin ce A
In
e
um
mp l
si
v
R e so l
u
1
+2
x
u
n
x
migh t
eq
a
3
2
A ss
my
B
11
5x
.
we
11 ;
A = 3,
m
e,
2 A + B = 5,
en ce
E x a p le 2
3) + B
n
+b
a
+n
n
M—n
x
+b
o
fA
y
an
v u
d B , b u t it
al e
we p lease
.
HI
E
xa
A ss
wh
be
m
pl
e
v
R eso l
.
9x
e
um
9
e
A is s o
in e
e ter
u
(
2
2 4x + 4 8x
an
d f ( x)
a
h
u
2 4 x 2 + 4 8x = A
3
an
(x + 1 ) f ( )
x =
mi
9
(x
n c tio n
o
f
wh v u
x
o se
al e re
mi
a n s
to
3
(
(
x
x
+
1 6x + 1 6 ;
th e p artial frac tio n
ne
x
x
3
2
1
2
2
12
24
3
4
Z
Z
—2
+ 6z
6
1
x
3
2
+
d
in g to
x
4
12
24
(x
(x
(x
pu t
+ 1 2z + 2 4
z
6
z
.
4
2
.
16
s c o rre s p o n
+ 16
(U
9x
(
x
3
x
f( )
e te r
+1
d tran s p o s in g,
x
d
.
A
en
S b s tit tin g fo r A
To
u
f
s
f (x )
A
x
t,
.
.
1, t
x
+ 1)
x
x
G E BR A
to p arti al frac ti o n
in
x
e c o n s tan
9x
L et
AL
24x 2 + 4 8x
3
3
x
(
m
d md
e re
(
GH E R
(x
4 8 3:
3
m
m
m
um
h
e ra t o r h as b e e n
I n all t h e p re c e din g e x a p le s t h e n
in a t o r ; if t is is n o t t h e c a se ,
o f lo
e n s io n s t an t h e de n o
e r di
in a to r n t il a re a in de r is
w e di ide t h e n
e ra t o r b y t h e de n o
is O f lo e r di e n s io n s t h a n t h e de n o in a t o r
O b ta in e d w ic
318
w
v
E
xa
m
.
h h
m
pl
um
h
v 22I21:
2
3
R es o l
e.
v
m
w
B y di is io n
e
2
7
1
u
in t o p artial frac tio n s
3x
1
1
let
—
1
x
f
f
.
3
5
.
.
,
2x
We
319
ra c t io n s
ra ct io n in
m
m
h
al
l
w
l
a in
h
w
o
p
a
b
e
sed t o
a c ilit at e t h e
y
a sc en din
w
rs o f x
o
e
gp
m
s
u
n o
ex
f
.
i1
u
l t io n
e x p an s io n
res o
in t o
of a
p
a rtia l
ra t io n a l
PA R TI A L FR A C TI O N S
E
xa
s eries o
m
pl
e
1
d
f a s c en
mf
d
Fin
.
th e gen eral ter
w
in g p o
e rs o
(
x
_
— 2x
3
)
(
x _
1
é
‘l
m f th
H e n c e th e ge n eral te r
xa
m
pl
e
2
m
th e ge n e ral te r
A ss
d
E x p an
.
O
"
2 33)
5
_1
2x ’
+
6
(
2"
5
1
F
6
2
2
2)
—x 2
2
)
(
)
w
1
(
‘ ‘
2
x
1
)
2
5
r
7 +x
in
( 1 + x)
as c e n
d
w
in g p o
ers
o
f
x an
d fin d
.
um
A
_
e
l +x
Le t 1 + x = 0,
+
Bx + C
‘
1 +x
atin
g th e
ab s o
l te ter
eq
atin
g th e
fi c ien
co ef
7 4
-
x
ts
7
s,
f x9 ,
o
)
l
wh
wh
C,
A
:
O= A + B ,
en c e
C: 4 ;
en c e
B
3
.
4 — 3x
3
11:
’
2
th e n A = 3 ;
m
u
eq
1+x
1+x
2
3x ) ( l + x
3
'
— x + x2
— 3x
)
fi c ien
c o ef
To fi n d th e
1
( )
h
t
ere fo re
2
( )
an
If
d th e
r
v
is
e en
in th e
If
r
re q
is
o
fi c ien t
c o ef
th e
dd, th e
{
2
—
1
x +
th e
"
of x
in th e
v
e
"
fi c ien t o f x in th e
co e f
s ec o n
r+l
fi c ien
c o ef
in to part ial
7x
s ec o n
"
fi cien t o f x i s 3 + 4
co e f
t is 3
1)
2
E XA M PLE S
R es o l
4
x
"
t o fx
ex p an sio n
u d
ire
,
l
a
is
e ex p an s io n
7 +x
1
( +
in
4
—x
-
u
u
(
2)
x _
5
—
E
e
4
5
1
'
dd
.
1
2x )
-
en e xp an
a e
.
3x 2 + x — 2
fx
wh
o
hv
By E x 4 , A rt 3 1 6 , we
.
26 5
.
3
.
d
d
s eries
is 4
5
1)
.
s e rie s
m
+2
2
.
XXIII
.
f
rac tio n s
46
(
I) ?
is — 3
l 3x
1
3
x
26 6
HI
x
(
z
AL
GEBR A
3
2x
13
1)
x
GH E R
(
x
2
— 5x
+ 6)
9
<
(
x
x
.
+
— 3 x3 — 3x 2 + 1 0
u
(m
2x
3 x3
(
Fi
n
8x + 1 0
d th g
di g p w
e n eral
e
as ce n
(
x
o
n
m f th
te r
ers o
o
fx
x
+
se
2
)
1 1x + 5
2
6x
1)
2
fo llo win g
e
—3
x
3
5x
2
~
—x — 3
x
4
x
(
x
2
+ 5x
(
x
+
wh
e xpre s s io n s
en
l 1 x + 2 8x
(1 + x)
16
2x )
2x
3
x
.
2
( + 3x )
2
1—x
WH Y
1)
(
23
ax
— bx
1
(
)
)
Fin d th
.
e su
(1 + )
x
2
‘
2
—
—
l 2x
3
—
l
x
)
2
2x
3
2 2
—
(2 3x + x )
I
—
1
(
+3
4 + 7x
2x + 1
(x
7x
4 + 3 x l 2x
2
—
—
( 1 x ) ( 1 + x 2x )
2
4x)
1
(
+
2
-
-
2
—
(1 x ) (1
2
2
—4
2x
de
.
x
1
ex p a n
l
(
mf
o
n
ax
)
)
ex
m
t er
s o
f th e
s e rie s
1
( )
x
2
( )
24
W
.
h
1
(
1 , fi n d th e
en x
su
1
<
1—
Su
25
.
x
mt
)
<
—
l
m th
te r
s
Si o n s
Pro
h h
w
ic
v th
e
can
-
at
th e
e
in
o
n
ite
>
<
— x6
1
h
w
o se
(1
1
p
“1
e
o
f th e letters
x
)
s eries
4
>
< >
1 — x7
+
mi
t er
s
)
— xp
1
(
1
(
o
fi
x
m f th h mg
md
b e fo r
su
3
e
2
e s eries
— xp
1
(
)
.
o
w
0
2
x
26
m f th
x
o n
(1
ax
z
a
o
+2
e n eo
a,
b,
u
)
s
c an
p ro
dt
u
d
c ts o f n
hi pw
e r
o
e rs
m
di
is
en
d
2 68
HI
m
w
GH E R
AL
GE B R A
mth
.
m
u
f p ro c e d re is t h e s a e
a
c o n s is t o f, t h e
an
te r s t h e s c a le o f relat io n
o
e er
y
o llo win g ill s t ra t io n will b e s f c ie n t
t er
kn
a re
s
o
h wv my
f
u
n
A
th e
s
m
.
l — p x — qx —
s c a le o
f re la tio n
hv
a
o
n
_
_
x
p
u
=
a
co e
y
an
ffi i
c en
t
a x
.
v
v
m
E x a p le
3
m
th e
,
d
fu
wh
eq
u
en c e
13
atio n s
a
+ rx
3
a
.
-
n
3
x
n
—3
ra
wh
s
u ffi i
f relatio n
2 g= 0,
5p
hu
6, t
t
c en
c o e ffi c ie n
th e
en
s
th e
1
an
u mb
n
er
t
s
o
f th e
f th e
o
h
z
qx ; t
d 35
sc ale o
u
rrin
to
o
rec
en
1 3p
o
o
f
a
.
g serie s
b tain p
5q
s
an
d q we
hv
a e
O;
f relatio n is
6x
5x
e
a
m
f th e te r
n d
o
my b f u
f re la t io n
o
s cale o
=
=
5
n
d
a
,
p
q
—2
.
if
,
s c a le
th e
n
a — x
n
2
.
be o nd
a re k n o wn
y
e rs e l
Fin
.
s
2
L et th e s c ale o f relatio n b e 1 — p x
th e
3
a _
q n 2
can
t h re e p re c e din g t e r
32 3
Co n
s e rie s b e gi e n
q
x
a
p
n
s
+
a _ x
n
1
.
o r
hu
2
2
l
rx
i'
e
a x
t
a x
d x
m
s e rie s
f th e
o
o
.
z
a
we
d
o
ufi
If
is t h e
e
2
m
v v
I f t h e s c ale o f re la t io n c o n s is t s O f 3 t e r s it in o l e s
3 24
u st
a e 2 e
a t io n s t o
e
d
2 c o n st a n t s , p an d q ; a n d w e
q
To o b t a in t h e
rs t o f t e s e we
s t kn o
t e r in e p a n d q
s
O f t h e s e rie s , a n d t o
O b t a in
t h e s ec o n d we
a t le a s t 3 t e r
i
a e on e
o re t e r
en
T
s
a
s
a
l
f
st
t
o
o
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t
i
n
a
c
o
e
g
in o l in g t w o c o n s t a n t s w e
st
s
a e a t le a s t 4 t e r
re la t io n
m
.
.
m
mu h v
v
g
i
en
m
-
m
v v
hv
fi
.
m v
.
-
u
h
mu
w
hu
mu h v
m
.
s c ale
I f th e
re la t io n
Of
b
1
g
fi
n d th e
q
st
a e 3 e
a t io n s
T
a in t h e
rs t O f
o
o
b
t
q
w a t le a s t 4 t e r s o f t h e s e rie s , a n d t o o b t ain
a e t wo
st
o re t e r
s
i
n c e to fi n d
e
n
e
;
g
in o lvi n g 3 c o n s t a n t s , a t le a s t 6 t e r s o f t h e
e
px
x
rx
to
s w mu h v
u
fi
m
w m
utk
th
m
m v h
h t w w mu h v
th
l ti
l
f
v
m
u t b giv
i s m
lly t fi d
s l f l ti i v lvi g m t t
G
w m
u t k w t l t 2m
u tiv t m
v ly if 2m uti t m giv w my um
C
f l ti
f th s l
3
c o n s tan
a
t
e
s
n o
er
o
e
o
re a
s ca e
e n e ra
or
s
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on
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e
on
en
e
s
s er e
e
.
e
ese
e o
t
o
,
ca e o
,
a
.
n
a
eas
ca e
re a
1
re a
c o n sec
ve
co n sec
,
o
e
er
on
pl
x
p gx
2
p ax
on
3
er
s a re
n
s
c o n s an
n
o
s,
.
en
,
e
a
a ss
e
R EC
To
3 25
.
mt h
Th e
s c a le O f
t wo
onl
fi
e
d th e
n
c o n s t an
L et th e
ts
s e rie s
y
269
.
a rec u rri n
s o
s
s
h
sa
e
al
l
s
s eries
g
m wh
at e
e
u pp
v
.
it t o
o se
b
er
th e
e
co n
t a in
.
be
mb
+ a x + a x
0
2
2
1
+ a x
3
1
( )
3
S ; le t t h e s c a le
re a t e r t
al e O f n
g
d le t t h e s u
S O t a t fo r e e r
h
mf
mi t h
ter
n
m
a
an
o
fi
o
re la t io n
y
mf
su
S E R IE S
f n din g t h e s u
we
; fo r s i p lic it
d
o
UR R IN G
e
v yv u
h
o
f
re la t io n
be
hv
1, we
an
2
—
l p x —q r;
a
e
0
N
o
w S
x
q
l
(
—
+
0
”
l
p
a
o
q
x
S
)
g
fi
a
fe
o
(
0
S
hu
n o
mi
n a to r
3 26
.
de c re as e s
su
_
“I _
mf
o
is t h e
a
—
1
q
p
2
0
o
er
x
re c
a
a
u
f
y
p
o
(p
w
(p
a
n
a
er o
g
)
x
in
z e ro
q
a —
g it 2
—l
s e ries
x
)
is
a
q
c o n se
en c e
Q
m
(L
f
a
ra c t io n
wh
o se
u
in t h e re s lt o f t h e las t
in c re a s e s in de n it e l , t h e s u
h f
u
q
de
.
ra c t io n
n
l
r
x
°
fi
1
v
a
"
2
—
—
l px
x
q
rrin
,
x
a
q n _2
f x is
x
"
0
g a —2
f re la t io n
I f th e s e c o n d
in de n it e l a s
)
x
al
a —
n
l
)
l
r
a
t
o
p
)a
s c a le o
fi
2
o
—
l p x — qx
th e
s
(
a x
er
n
O
pa
v y h
a
a
2
l
q
fo r t h e c o e f c ie n t
o f t h e re la t io n
T
a x
p
a x
S
x
p
+
d x
S
x
p
= a
—
p
m
y
—
x
a rt ic le
q
o
f
an
x
,
w
I f we de e lo p t is ra c t io n in a s c e n din g p o e rs o f x a s
w
e
s
a
ll
a
t
h
A
rt
3
1
4
O
b
t
a
i
n
a
n
t
e
r
s
O
f
e
e x la in e d in
s
,
y
p
fo r t is re a s o n t h e e x p re s s io n
o rigin a l s e rie s a s w e p le a s e
h
.
a’
c a lle d
3 27
t h e gen
F mt h e
ro
.
“
0
era tin
+
l
(
a
_
1
p
x
pa
0
x
q
)
2
px
gf
res
h
o
1
is
u n c ti o n
u lt
o
2
qx
o f th e
0
+
s e rie s
f A rt 32 5, we
.
x
a
m
a x
1
+ a 2x
2
+
l
o
.
b t a in
+ a
px
x
x
q
2
"“
m
270
HI
f mwh i h w
c
ro
h
t
e se e
at a
GH E R
lt
a
h u gh th
(
s
it
u
be
a
to
y if th
l
.
x
x
p
q
u
e re
mi
a n
m
0
u
+ a x + a x
2
+
2
1
a
an .)
1
x
p
.
x
"
es
en
e s e r es
en
o
a
e
a
x
q
o se
en
A
a
{
h
mh
to
d
l
wh
s c ale o
m
'
'
l) Bb
mf
o
p
'
(
r +
1 ) Cc
ms my b
a
e
n c tio n
d
en o
te th e s u
-
h
2
qx ; t
px
d th e
s c ale o
en
m f th
O
f relatio n is
6x
x
e s eries
xS
x
6x 8
ic
u
is th e ge n e ratin g f
S = 1 — 8x ,
n c tio n
.
.
h
2
2
—
—
6
x
)
(1 x
2
; t en
S = 1 —7x
wh h
y
l
re ss e d as
p
re c
u
rrin
g
fu
g
g
2
a
s e rie s
n c t io n
°
'
}
x
f
o
”
.
u
n
d
wi h ut u i
g
m
m
t
th e gen eral t er
,
2 —
—
—
1 7x x
4 3 x3
1
L et S
ex
e n e ra t in
— cx
l
(
)
t er
n
u
u
an
on
o
s n
.
f relatio n b e 1
1 , q: 6 ;
o rds
O
1 + 7p — q = 0,
en c e
er
’
bx
+
Fin th e gen eratin g f
E x a p le
s o f th e rec rrin g s erie s
n ter
L e t th e
h w
s
.
.
t
be
Of a
th e
B
I n t is c a s e t h e s u
th e
e t o d O f A rt 3 2 5
m
m
o
c an
t er
o se
mi
t h e gen e ra l t e r
in
ra c t io n s
a rt a
e
o
l — ax
h
s
s,
.
n
e n e ra l
e
A
T
u n c ti o n
en era t n
n
o
2
.
ra c t o n s
eas
“
n c re a s e d
e
n
e r en
e
a rt a
ec o
e
c an
e
n
s co n
en
.
ro
s
n
at
s
v i h wh
i i d fi it ly i
wh th i i v g t
Wh
3 28
th
i
g
gf
i
l
i
h
u
f
f
t
g
p
p
g
ily f u d
u pp
my b
Thu
mp d i t th p i l f
b d
s
fi
de r
(p
an
n c t io n
96
05
my
o
a
o n
fu
g
n e ra t in
e
g
e
b t a in a s
an
t e r s o f t h e s e rie s a s we plea s e ,
n it e s e rie s
a le n t o f t h e in
i
re a rde d a s t h e t r e e
v
g
q
se d
be
c an
.
)
mm
o
l
my
GE BR A
AL
2
— 4 3x 3
7x
2
3
6x
2
x
x
4 2x
3
,
an
d th e
su
27 2
HI
GH ER
AL
G E BRA
E XA M PLE S
XXI V
.
Fin
.
s e ri e s :
1
.
3
.
d
u
t h e ge n e ratin g f
n c tio n
.
.
m f th
d th e gen eral te r
an
2
4
.
2
.
7
— x
2
5
x
+
o
-
7x
3
fo llo
e
wi g
n
+
5
.
Fin
6
.
8
.
9
,
d
th e
n
m
th
te r
an
d th e
su
mt
o
m
ter
n
7
s o
f t h e fo llo
wi g
s e rie s
n
.
l
3
11
S
.
hw h
t
e
th e
at
s e rie s
+
71
+n
a re re c
c
u
u
rrin
g
s e ries , an
hw
d du
h o w to
S e
12
rrin g s e ries
fro
.
mth
13
.
e su
Fin
mt
d
d fin d t
e
a
o
in fi n ity
th e
su
e r s c ales o
th e
ce
a x
o
hi
su
a 2x
l
o
.
Th e
o
1 + p x + qx
gen e ral t er is
m
are
,
(
1 +
an
-
i
.
fi rs t
e
n
m
te r
s
o
f th e
re
m
+ 1 t er
n
s o
f th e
s e rie s
— 53 +
f th e
b0
2
,
.
m f2
s c ale s o
3
3
a x
3
3
14
,
f re lat io n
m f th
2
2
rx
re c
bl
+ sx
x
2
,
u
rrin
g
s eries
b z xz
b 3 x3
v
res p ec t i e ly
" s a
re c
x i
u
rrin
g
;
S
h
e
s e rie s
w t
wh
h
at
o se
h
th e s eries w
s c ale is
o se
1
md h v
a in g fo r its
1 5 If a s eries b e fo r e
t at it
o f a gi e n re c rrin g s e ries , s e
o s e s c ale o f relatio n
ill c o n s is t
s e ries
o f th e give n s eries
.
v
wh
u
hw h
w
.
n
th
will
o
f
mth m f t m
l
u ig
f m
m t mth th t
ter
a so
one
o
e su
or
o re
a
er
n
rec
an
er
s
rr n
a
XXV
CH A PTE R
C O N TI N
331
A
.
e x p re ss io n
n
o
UE D
.
FR A C TI O N S
.
f m
f th e
or
I
a
c;
e
h
u ed
frac t io n ; ere t h e let t e rs a , b , 0 ,
re s en t w e
a te e r, b u t fo r t h e
h
an t itie s
p
q
1
w
e re a , , a ,
a
th e S i ple r o r
l
2
1
c o n t in
w v
m f m
u
in t ege rs
sh
h
Z+
u su lly
hs
T i will b e
.
a
w itt
r
a
de n o t e a n y
a ll o n l
c o n s ide r
a
y
o
s
i
i
v
t
e
p
a re
a
m
o re
co
mp
act
f m
or
a3
2
um
h
in t h e
my
d
1
1
a
en
c a lle
is
fi
b e r o f qu o tien ts a ] , a 2 , a
is
n it e t h e
W en th e n
3 32
in a tin g ; if t h e n u b e r o f
ra c t io n is s a id t o b e ter
c o n tin
ed
ra c t io n is c alle d an in
i
n li
t
ed th e
i
n ite c o n tin u ed
n t s is
o
t
e
q
u f
.
f
u
ra c tio n
u m
m
f
.
fi
v y t mi ti g
mplifyi g t h f ti
u
I t is p o s s ib le t o re d c e e
rac t io n b y s i
t o a n o rdin a r
t h e lo e s t
b egin n in g ro
yf
f m
333
Let
u
q
To
.
ni
t ie n t
c o n vert a
b e t h e gi
—
n
o
w
an
d p th e
re
en
mi
a n
er
er
f
cti o n in
ra
to
ra c t io n
i
i
d
e
d
;
de r
t
v
hu
2
n
.
H A
.
n
e
n
?
d
n a
co n
ra c
u ed f
i su
tin
ons
ra c t io n
s
c c e s io n
n
.
i
v
n
e
g
v f
m
'
s
a co n
tin u
mb y
n
,
ed
frac tio n
le t
a
l
.
b e th e
27 4
HI
v
di ide
th e q
by p,
n
G H ER
u
AL
G E BR A
t ie n t
o
.
d q th e
an
n
1
)
17
1
re
mi
de r
mi
de r ;
a n
q
v
di ide p b y q ,
T s
on
th e q
hu
.
If
mi
s
le s s t
h
an
n
,
the
fi
u
o
t ie n t
rs t
u
q
m
d p ro c e e d
w ill b
be
as
f
o re
v
7
th e
t ie n t is
re
z e ro
,
a n
an
an
d
SO
d we p u t
1
n
n
an
o
d
an
°
.
dt
h
fi
e O
xa
v
m
pl
d
e
mm
wh
v yf
s
E
b se r
e.
h
Re
du
we
a
mm m
um
v
78
2
Fin in g th e greates t
r
o
s
c
s
e
,
p
th e
e
h
2 51
ce
at
bo
v
m
h
i
t
h
a
a s t at
s
s
e
e
p
on
an d n
h
re a t e s t c o
e a s re O f
h
n din
t
e
e n c e if
o f
g
g
a re c o
en s u ra b le w e s a ll a t le n gt h a rri e a t a s ta e
an d n
g
T s
e re t h e di is io n is e x a c t a n d t h e p ro c e s s t e r i na t e s
w o se n
e ra t o r a n d de n o
in a t o r a re p o s it i e
ra c tio n
e er
in t ege r c an b e c o n e rt e d in t o a t e r in at in g c o n tin e d rac tio n
It
to
co
hv
m
u
ud
m u
tin
mm
on
e as
a e
o
re
8 02
3
6
49
8
m
hu
v
.
f
.
.
f 2 51
2 51
6
m
u
fractio n
e
v
m
m
a co n
5
ro c e s s
an
d 8 02 b y th e
uu
s
al
1
an
d th e
s
u
v u
c c es s i e
q
o
tien ts
are
2 51
8 02
3 , 5, 8 , 6 ;
1
1
3+
en c e
1
5+
f
h
8+
1
6
°
fi
Th e ra c tio n s o b ta in e d b y s t o p p in g a t t h e rs t, s e c o n d,
3 34
ed
ra c t io n a re c a lle d t h e
o t ie n t s o f a c o n t in
t ird,
rs t ,
q
c o n vergen t , b e c a s e
s ec o n d, t ird,
a s will b e
in
s e
n
,
A rt 3 3 9 , e ac h s c c e s s i e c o n e rge n t is a n e a re r app ro x i a t io n
t o t h e t r e va l e o f t h e c o n tin e d ra c t io n t a n an y o f t h e
re c e din
co n
er en t s
g
g
p
.
h
.
h
u
u
.
v
u
v
u
.
v
s
u f
u
u
f
hw
h
fi
m
27 6
IGHE
H
(
Th e
th e q
+
n
uo tie
1
t
n
v
co n
e r en
g
G
R A L E BR A
n
f mth
f
t di fe rs
in t h e pla c e
a
.
ro
o
f
e
n
1311
on
h en c e t h e
a
y
l
(
h avi
in
n
1)
n +
th
g
co n
t
v e rgen
(
a
n
q
n
-
g
l
2
it -
)
g
n
—l
n
+
a rt
“
n
I f t h e re
an
t h at t h e
see
—I
by
s
+l
ume
p
+
1t
o
su
3 37
th e
We
co
s y
u
m
l
t
q
p
sh
e e
l
al
We
u su
h ave
o t ien
yd
a
ll
s
ee n
en o
h s st
t e th e
on
yi
l
n
9 1.
i
nt
o
t
e
q
u
u ed
f
hu
3 t
3 38
If 2
.
er
an
co
e e
q
n
h
h
h f
th e
o
n
th
an
u
p a rtia l q o t ien t ;
1
1
au
tie n t
+i
at an
+
an
y
s ta
+2
e
g
by k
.
2
g —l
g
n
it
—2
b e de n o te d b y
u
q
e e
o
x
t
;
h
en
a:
t ie n t k in s te a d
o
f
di fe rs
f m
ro
f t h e p a rt ial
s
b e th e
n
th
n
t in
u ed f
71
q
—1
n
c o n ver en
g
pn q
co n
rc
+ pn—
mpl t
qn
Le t t h e
g
th
b e in g
e
g
mpl t qu
co
k9
9
a
“71 29
ra c t io n
t ak in g t h e
an
+1
t h at
;
t in
m
c a ll a n
t to
t i
at
t
gr
co n
+1
+
d de n o in a t o r o f t h e ( n
1) co n
wa s s pp o s e d t o o ld in th e c a s e o f
o ld in t h e c a se o f t h e t ird c o n
t h e o rt , an d s o o n
t e re o re it
c o n v en ie n
p
let t h e
n
“n
.
I t will b e
.
.
wh h
u
sh
s
fu h
e
.
v
q
p n —1 ’
ic
law
th
B u t t h e la w do e
th e n
fo r
e rge n t , h e n c e it h o ld
n ive r all
h o ld
o
o s itio n
ra t o r a n
n
f ll w th
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XXVI
C H A PTE R
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h
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v
34 5
I n C ap X w e h a e s e wn h o w t o o b t a in t h e p o s it i e
it n
in t e gra l s o l t io n s o f in de t e r in a te e q a t io n s
e ric a l c o
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e ffi c ie n t s
t o Ob t a in t h e ge n e ral o l t io n o f a n y in de t e r in at e e q at io n o f
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a an
34 7
equ a ti o n
th e
a
e
,
s
er
co n s
o es n o
c
0 c an
:
o r a ge d :
so
ac o r
e a
o n s ax +
a
an
t
h
u m
m u
.
I t is c le a r t a t t h e e q at io n a x
by
in t e gra l s o l t io n an d t a t t h e e q at io n ax
t o by
ill b e s ffic ie n t t o
a cc
0
e n c e it
a
by
t
z
tin
h
g;
5
t
in p o s itive in tegers
th e
.
ued f
en
f
o
ra c tio n
aq
bp
,
an
i1
d le t
.
?
de n o t e
rt
A
[
.
IN
I
D
ETE R M I N A TE
I f aq
.
v
l , t h e gi
bp
a
(
a
N o w s in c e
di is ib le b y b
h
en
w— b y
O F TH E
u
q
e
hv
a
e n c e a:
—
E
r
e
R EE
285
.
en
.
mm f
wh
b t,
a
DG
w itt
bp ) ;
—
a
co
e no
cq
se
= 0
FI R sT
my b
a t io n
( q
—c
—
= b
c
y
q)
p)
(
x
d b
an
a
v
U A TIO N S
EQ
on
t is
e re
mu t
w — cq
a c t o r,
s
in te ge r,
an
be
cq
b
t
h
at
is ,
x =
f mwh i h p
ro
c
to t
an
um i
y p
v
o s it i
l
t
a so
II
e
v
er
a
my b
a
u s th
x
to t
an
t wo q
y p
u
an
III
.
v
o s iti
t itie s
If
v
o s iti
e
b
h
y
a
(
029
b t — c q, y
er a o r
b is
u
n
.
a
ed
.
r) ;
t,
—
an
t e ge r
1n
= at — c
p
;
u
my b
wh h
d
e n
n
p
b
u lim
it
s
c
—b
aq
“
v u
t hus th
cp
cq
e it
u
in t e gra l s o l t io n s
in te gra l a l e
ic
e
t it ie s
an
f s o l t io n
cq
e
—C
=
2
z
c
v
v u
v
u
q
si
f t h e t wo
er o
a
by
-
o
um
b
hv
Cg
w
h en c e
ro
u
e n
— l we
,
z
dx
f mwh i h p
i
m
u
t h an t h e les s
h
t
;
e z e ro
I f a q — bp
.
s
= at + c
p
in t e gral o l tio n s
a
b
e o b t a in e d b
i
i
n
y
y g
g
in te gra l a l e , o r an y n e ga t i e in t e gra l a l e
e
y smll
e r c a ll
n
v
o s it i
b t + cq , y
it
ex c ee
u mb e
r o
y th f
w ith u
hw
,
f
s
so
u
l t io n s is
ra c t io n
e
v
b t a in e d b y gi in g
t h e gre a te r o f t h e
e o
a
1
u limited
n
(
c an n o
b
.
t be
co n
v tdi t
it
um t s d th
ti u d f
ti
i v tig t i
f il I t h
h
lu i
my b
w itt d w b y i p ti ; t hu if b 1 th qu ti b m
—
h
b
d
wh
d
t
l
u
i
s
m
b
f
u
t
;
y
y
y
y
ib i g t
i
t
i
v
i
l
v
l
u
h
t
t
t
y p
g
g
m
N
I
h
u
d
v
d
h
v
lu
d
b
w
wh h
mi
i
mm d
v
p
er e
n
o
es
a
on
n
o
en
r
ar
r
:
c
a
co n
a
n
s
e
n
e s e c a s es,
n
.
n s
en c e
ec
on
ra c
on
= ax
o
t
,
t
e so
e
e so
c, an
e ra o r
n
e ve r,
=
s
n
e
on
a
ons
on
e
a
an
,
o
e
a
e
eco
es
n
c
asc r
n
o so an
os
e
n
e
ra
a
e
re a e r
an
a
OTE .
t
o arith r
res
ec ti ely .
t s o l b e o ser e
t cal p ro gres s o n s in
t
at
ic
th e
th e
s eries o
co
f
on
a
es
fo r a: an
ifferen ces
are
or
f
y
b an d a
28 6
HI
m
E x a p le
d
Fin
.
th e ge n
v
g
g
w h v h
In
ertin
co n
g
in to
29
29
co
mb i
n in
u
ud
tin
G E BR A
v
h
h
v
g t is wi t
th e gi
13 —42
x
6 5— 4 2
x
eq
en
th e gen
u
x
atio n
9
— 1
;
45
— 5
;
x
we
,
+ 65
by
c,
to
fi
L et h , h b e
u
= 42t
d th e gen
n
a so
u
l tio n
x
t h e ge n
.
Let
;
c o n ve r e n
g
I
.
To
fi
f ax — b y
o
by
t
st
b e fo re
33
n
e ra l s o
co n
(
—
h
:
u
l t io n
ju t p
s
I f a q — bp
in t o
e rt e d
re c e din
=
h
a
g
(
c
cq
z
h
en
ah
— bh
—h
t,
e
x
z
c
= le + a t
.
;
lu tio n in p o s i tive in tegers
by
G
so
z
t in
3 th en
v
by
.
u
ed
f
ra c t io n
bp
aq
=
l= 1
,
an
.
e
= c
<q
a
bp ) ;
—
+ cp
y
a
bq — b t y
,
z
t,
at
an
1n
—c
p ;
t e ge r
d le t
i
u at o n
q
in te ge r ;
an
q
—
th e
.
b
x
f
o
h — bh ;
y
5
+
ax
t
a
a co n
l , we ha
a
.
c
z
a
z
b t, y
+
d th e gen era l
v
45
x
ax
be
u
.
.
b
349
equ a tio n
j
t
4 2y = 5
in te ger
an
i n p o sitive in tegers
era l s o lu tio n
x
s
ergen
tio n
s o lu
one
“
c
t,
6 5, y = 2 9 t
ax —
wh i h i
v
f 29x
l tio n is
eral s o
Given
.
co n
o
b tain
o
y + 45
29
42
x
3 48
in tegers
(y
x
e n ce
e
frac tio n th e
e
29
h
.
l tio n in p o siti
e ral s o
a co n
AL
erefo re
t
a e
e
—
GH E R
z
3
q
28 8
I
Le t
.
1
bp
aq
t
Po s iti
t
o s it i
h
v
v
in t e gra l
e
be
rt 3 4 9
A
[
.
]
.
v
t a in e d b y gi in g t o t
09
an d
t an
n ot
le
’
Ob
h
g
u
S pp
t
o se
h
at
2 a n d __ a re
c
b
a
ss
b
f; z
i
t iv i t g
h m
p
v lu
h v i m
f th u mb f lu i
th
w e re
,
t h e le as t
a re
n
os
t
e
a
h
N o w t is is
f
ab
ra c t i o n
u mb
er o
—
din g a s
h
I n t is
h
t is , t h e
c
b
n
H
1
c as e
g
er
th e
ab
n
n
S pp o s e t
u mb
er o
f
f
g
c
b
f
e
+
v lu
f
1
le s s t
n e a res t
e
g
a
g
a
;
.
ritt e n
re a t e r o r
IS
ra c t io n s
d t h e gre a t e s t
d
a
b
h
an
0
to
7
g
.
ra c t io n
Th
or
,
us th
re a t e r o r
g
,
(t
f
a
e
le s s
an
d
in t ege r
an
.
on e
a
u
u mb
er o
f
h t 2i
s an
a
v u
u tio
n
s
f
e o
is
a;
z e ro
i
h m
t b
wh
u
,
j
b
lu t i s i t h g
t t i t g
lu d
lu d t h
lu t
i
+
u
If we in c l de
.
c
s
e
in
an
a
0, an d o n e
a l e o f t is
so l
v lu
s o l t io n s is
dI n g a s we
h
—
f
, g p ro p e r
d my b w
ac c o r
I n t is c a s e
t is , t h e le a s t
th e
of
u
i
i
i
( )
h
O,
c
or
—
f
+
g,
.
u mb
en c e
%
7
an
.
i
s th e
re a t e r
g
g
or
u
( )
1,
is
is t h e in t ege r
ons
f
g
on s
an
as
is
S pp o s e t h a t
b
11
.
d
ac c o r i n
o
ac co r
t e ge r
i t ege r,
n
an
,
q——
t
n
dj
an
+
s
so
m
s lu ti
f
e
a
e rs
e
n
er o
n
a
e
c an
e n
e re o re
0
t in t e ge rs
n o
sq
Let
a
cp
re a t e r
t
n o
lutio n is
e ra l s o
at
s w ill
.
an
i
()
n
GE BR A
b t, y
u
e
a
AL
t h e ge n
en
l tio n
so
v lu s
in t e gral
e
h
cq
x
p
GH E R
H I
is
n
so
on
s
e
e o r ex c
In c
re a e s
e
n
e
er
e z e ro s o
in
ion
.
in t e ge r
.
v lu
a
u
I f we in c l de
f y is z e ro
re at e t is n ;
n
e
an d t h e
e
c
g
e o
m
m
+
1,
.
C
or
db
_
s
g+
1'
h
Th u s
th e
IN
n
D
E TE R M I N A TE
um
b
f
er o
c o rdin
g
so
u
l tio n s
u
u
S pp o s e t
h
U A TI ON S
ud
o r e x cl
e
h t 2 d?
a
th e
a re
an
in
bot
hi
n
f
In
ab
u
l t io n
so
E
0
t e ge r
z e ro
D
FIR S T
O F TH E
t h e gre a t e s t
IS
we in c l de
as
i
v
( )
EQ
t e ge rs
+
GR E E
1
u
c
b
a
.
1
.
II
.
I f we
ex c
I f aq
bp
u h
h
lu d t h
e
e z e ro
um
v lu
a
v u
th e
es
.
hv
hv i m
u
u mb
n
o r
.
I n t is c a s e
0 an d g = O, a n d b o t h a: a n d y
al e
I f we in c l de t e s e , t h e le a s t a l e t c a n
a
b e r o f s o l t io n s is n
t h e gre a t e s t is n
en ce th e n
v
28 9
.
er o
a
e
e
f
so
a
s
m
z e ro
an
,
1,
u
l tio n
d
o r
is
s
6
ab
an
mil
d si
a r re s
3 52
t io n
u
l , t h e ge n e ra l s o l t io n is
= c — at
x = bt — c ,
,
q y
p
ill b e o b t a in e d
ul w
ts
.
u
m
ax
f
:
u
f m
v u
wh h m
h v t w imu lt
I f we
.
a
ax +
m
e
o
by
+ cz =
b y e li in at in g o n e o f t h e
By
Ax
o f th e
o r
0
t e n t h e ge n e ral s o l t io n
f m
h
wh
e re s
is
.
x
u
h
u
S b s t it t in g fo r
s,
f
+
f
x,
or
H
.
v lu
v lu
H
.
es
es
a
A
.
e
a s a re a
e
s e
+
at o n s
ex
a n e
z
o
f
.
on s
a
w z
u
h
w
f
n
l'
'
v u
s =
a
a n
o
o
n
c
e
by
a
’
’
c z
z
d,
kn o n s,
s a , w e o b t a in a n
y
S pp o s e t a t x j , y = g is a
rit t en
can b e
u
d th e
in t e gra l
u
.
ro
’
a x
eq
B S)
y
:
g
z
_
A S)
.
u
an
’
d,
S b s t it t in g t es e a l e s
an
e
e q a t io n s , we Ob t a in
q
ic t h e ge n e ra l s o l t io n is
u
an eo
f th e
in t ege r
an
u
u
wh h
:
u
S
o
f mwh i h b y giv i g t
w
bt i
i
u
q
v d l dy pl i d
u q u ti
B y t ra n s p o s it io n a s: b y
d — cx ;
a l es 0, l , 2 , 3 ,
in s c c e s s io n t h e
’
ic
a
b
th e o r
ax
by = c ,
e so l
y
3 53
v
To fi n d t h e s o l t io n s in p o s it i e in t e ge rs
cl, w e
a
ro c e e d a s
by
ex
o llo ws
y p
.
.
o
h
we
+
u
f x
a t io n
o
Gt,
z =
o
b ta in
Bh
+
B Gt,
y
z
a re
,
y
o
d y in
o f th e
an
le — Ft
e it
h
v
f t h e gi
Fs + Gz = H ,
er o
f m
sa
y
or
en
o
f
.
— A h — A Gt
;
g
v
b t a in e d b y gi in g t o t
s
u it
ab
le
GH E R
HI
3 54
If o n
.
e so
b
fu
u
by
h
b
,g
Le t j
,
bg
+
t ra c t io n
(
wh
bc
u
l tio n
a
f
f
b
< > (IV
f)
fr —
—
+
b
+
f
x —
t is
e re
'
’
b
an
’
’
c
c, c a
f
'
—
(y
9)
g)
o
b
c
(
+
'
b
y
’
Fi
n
ab
c a
o
o
s
.
en
'
h
d
’
.
<
’
(
s
d
hu
’
th e gen eral
.
In h o w
wn s
Fi
n
d
so
u
l t io n
7 1 1y = 1
4
t
’
a
'
—c a
.
.
'
Ic
b
’
.
mywy
th e
n
a
er o
u
t
—
)
lc
XXVI
51 9y
4 55x
=
.
1
in tegral
e
3
.
u
v
=
1 03 1
’
b)
e
u ti
so l
4 36x
.
di
.
f s o l tio n s in p o s iti
1 1 x + 1 5y
n a t o rs
.
£1 1 9 3 6 d b e p ai
.
mi
— a
v
s c an
u mb
.
d th e le as t p o s iti
2
.
an
an
.
.
'
7 7 5x
h
in t e ge r a n d h is t h e H C F o f t h e de n o
’
T s t h e ge n e ra l s o l t io n is
a b
a, a b
.
1
5
n
39 3y
flo rin s
in tegers
o
on o
an
d
f
5
=
.
h lf
a
f
.
d w f i h vi g 7 d f th i d mi t
d
u h th h i mi 3g
Fi d w p p f i i h i l w
mh i g
7
d
mi t
f th i d
d u h h t t h i d iff
i
i
A
m i t f p u d y h illi g d i i h lf
8
hilli g fi d h m
fy p u d
S lv i p i iv i
g
6
.
Fi
n
at
c
an
8
t
n
.
or
o
o
t
o
s
t
ro
o
e
s x s
n
os
e
rac t o n s
s
co n s s s
s
n
;
te
9
an
or
e r
en o
n a o rs , an
.
er
su
n
t
1—
,
n
a
n a o rs an
en o
c ert a n
n
rac t o n s
e r su
e r
.
o
c
f ll w
as
z
z —
g
'
'
ca
c
—
.
s
bg
+ c
E XA M PLE S
c ro
'
+
h
t
d,
z
b t a in e d
e o
a t io n s
en c e
bc
wh
a
so
d,
ch
» w—f
'
my b
u
l t io n
qu
,
0
a
so
e
'
c z
u
f
su b
d,
z
f th e
o
'
t h e p a rt ic la r
e
a
By
+ cz
d, t h e ge n e ra l
n
o
e
.
l t io n in p o s it i ve in t e ge rs
ax +
c an
G EB R A
AL
n
t
o
e su
t
n
e r
es t
o
t er
av n
s
12
1
c
t
a
x
o
n
e r
s
s
e ren c e
n
s
24
t
s , an
.
ers
10
.
1 2 v — 1 1y + 4 z
— 4x
+ 5y + z
.
=
22
=
17
s
a
XXVII
C H A PTE R
REC
UR R IN G
UE D
C ON TI N
.
FR A C TI O
N
S
.
i C h p X XV
hv
h t mi i g
b
f ti wi h ti l qu ti t
du d t
y f ti w ith i t g l um t d d mi t
w
f
t b
u
l
u
d
b
h
ll
v
h
t
t
th
q
p
u
i
u
d
b
d
i
i
u
f
i
t
i
d
d
fi
p
q
id
wh qu ti t u W h ll fi t
um i
We
3 55
t in u e d
o rdin a r
.
a
c s
o
o se
ex a
mpl
e
n
c an n o
r
e
c an
s
en
e
a
e ex
re c
a
n
ra
on
ra c
ra
s een
t
on
ra c
e re o re
e
a
.
on a
e
o
a
r
s
u
as
an
S
a
co n
tin
e
.
s
c an
n
rs
t
n
n at n
ter
a
a
re
e
e ra o r an
n
re s se
r
en
o
ra
t
.
ce
en o
a
e
s
te
co n
co n s
er
o
.
an
n a o r, a n
ro
t
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a
n
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.
m
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frac tio n s ap p ro x i at in g to its al
m
.
v u
e
ud
frac tio n
e
,
an
d fin d
a
s e ri e s
.
3
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2
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1
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u
o
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t is th e q
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5
2
5
after
I
3
Q
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J
5
5
tien ts 2 , 1 , 3 , 1 , 2 , 8
re c
u h
r;
’
en c e
1
1
1
1
1
I
=4 +
1
9
J
2 §C 1 + 3 + 1 + 27 5 8 +
w
d h
wh h d u
t
ill b e n o tice
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o b le
s
i
i
c
i
en
t
o
t
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al ay s th e c as e
u
w
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o
u
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m
th e q o tie n t s rec r a s s o o n a s we c o e to
I n A rt 3 6 1 we S all p ro e t at t is
f th e fi rs t
at
.
.
h
a
ra c t o n
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n
at
e
d
v h h
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In
[ E xp la n a tio n
o
p e ratio n s
Fo r
.
h
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H
F4
i
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mp l
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in tege r in
greates t
UR R I N G
-
s
We
.
gate
co n
Th e fi rs t
so
a ratio n al
it
se
d
h
ltip ly
en
2,
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to
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t
en o
4
mi
9
d th e
ertin
in
n ato r
.
13
an
um
v
d
re
mi
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a n
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h
v h v
en
1
1
les s t
se
k
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or
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t
co n
3 56
ergen
1 02 4 00
t gi e s th e
v
an
o
as e x p la in e
d
4s
61
1 421
h
d
170
v u
al
fa
L e t x de n o t e t h e
an d s
t
at
o
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e
pp
u
h
lea s t fo
at
co n
t in
u df
e
ra c t io n
,
1
h
u
q
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p
p
9
o
h,
b,
,
9
t ie n t s h , h
hv
a
be
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re s
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2
,
u
q
o
t ie n t s
u
u
,
u
m
u
1
+
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be
;
v
e
an
co n
v
v ly
v re s e c t i
p
z
,
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g
h
wh h
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a
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an
an
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o
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ere fo re
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.
d im
als
ec
th e
s
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of th e
ra ti o n a l
one
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1
lc + y
1
to
h
hu
d is t
v
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1
y
in t e ge rs
e
a:
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c o rre s p o n
i
y s th e
co
.
din g t o t h e
mpl t
e e
qu
o
tie n t ,
—
p
to
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y
p
on
din g t o
the
'
r
en
m
ffi
x
ts
S b s t it t in g fo r y in t e r s o f
i
t
h
e co e
c ie n t s
a dra t ic o f
c
q
u
we b egin
—
x
q
p
e r en
t
e
r
d y t h e p e rio dic p a rt ,
+ v +
s in c e
en
e n ce
th e
ts
e rge n
v ly ; th
wh y
i
e
c
t
p
7
co n
s
at
‘
p laces
r
s it i
o
p
,
th e
f
h
ud
2, t
3
5
an
1
+
u
Le t
3
.
h
1
e re a ,
+4
E very p erio dic co n tin u e d frac tio n is equ a l to
u a dra ti c e u a tio n of wh i c h th e c o e ic ien ts a re
q
q
fi
.
ro o ts o
wh
J 1€
in A rt 33 6
les s t
fo rtio ri
to
e
h
is le s s t
e se
a
e s e rie s o
]
e
ft
m
in ato r b y th e
lt
res
326
Th e
er i s
en o
g th e
e sa
we fi rs t fi n d th e
d m
u
d
293
.
mth
md
ts fo r
e rgen
en c o n
FR A CTI ON S
E
e rato r a n
n
at a fter
t
NU D
v w
f th e lin e s ab o e e p e rfo r
lin e
c o n s i er th e s e co n
; t is is 2 ,
h
u
m
2
h
uj
J
wh
d
w
v v
—2
J1
C O N TI
y
y
r
'
s
y
x an
d
.
8
mplify i
Si
a re ra t io n a l
.
n
w
e
g
o
b t a in
a
29 4
HI
GH E R
u
AL
G E BR A
.
wh h v
'
v u
v v u
r
O
i
s
r
c
i
s th e
al e o f
Th e e q a t io n s y
e
,
y
g
(
)
re a l a n d o f o
i
t
i
n s
h
al e o f
a
s
r
s
o
s
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s
i
f
t
e
o
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t
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t
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i
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pp
g
p
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p
M
h
i
n ra t io n a lis in
t
e de n o
i
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n a:
b
s
i
t
d
o
e
b
s
t
t
e
g
g
9s
q
A + JB
e re A , B , G a re i n t e e rs ,
t h e a l e o f x is o f t h e o r
g
u
u
f m
t h v lu
v u
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E
xa
L et
wh
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h
v
e s in c e
v u
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u
al
b e th e
2
e
f th e
7=0
frac tio n i s
1
N/1 5
e q al to
ere fo re
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y
e
1
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tin
co n
.
eq
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1
v
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18
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t
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n
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24
.
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.
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1)
an
,
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t in
o
3
J 13
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7
.
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1
a
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frac tio n s ,
e
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.
3
.
an
d fin
d
th e
.
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.
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.
.
f th e
f th e
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wh
wh
rs t c o n vergen
26g
—
en
65
m
9 16
en
.
is
t to
t
k
en
fo r ,J 1 7
k
en
fo r J 2 3
ta
is
ta
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at
c o rrec t
.
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t o fi ve plac es
.
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o
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as
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.
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en
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.
t to
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h
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5
a
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2
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as
fractio n ; t
e
.
h
.
3+
E XA M PL E S
s ix t
i s re al
1
2+
al
wh
!
3+
e o
2x + 2 x
c o n tin
e
a
1
E x pres s
e.
x
en c e
is t
m
pl
m
’
.
as
a co n
e
at o n s :
2
—1
+ 2x
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d
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Fi
n
n
v g
udf
rs t co n
e
=
th e
a
a
22
.
ro o
e o
t
o
x
.
f
f3 +
e of
t to
J 15 t
x
2
1
1
2
—
h
at
is
v
th e p o s iti
rac t io n
e
O
h
v lu
v lu
eac
th e
tin
er e n
5x
3
1
1
1
e
ro o
23
4x
O as
1
t o fi ve p lac es
c o rre c t
a co n
t
Of eac
h
o
f th e
7 x — 8x
2
.
tin
u df
e
rac tio n
.
29 6
HI
We
S
h
a
c a ll
ll
th e
s e ries o
J
al
N
J
fi
t he
*
t it ie s
t
h
t e ge rs
9
Th e
h
F mt h e p
re c e din
o
+ a
r
a
a
+ as
mpl t
v
it
a
o
t ie n t s
s
3,
th e q
a
u
an
v
w p ro e
i
o
s
i
t
e in
p
a
a ls o
r
2,
.
ht
h ll
t
a rs
e
pp
in t e ge rs ; w e
e
r
4,
u
q
e e
a rt ic le
g
’
n o
v
.
p
t
p
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p
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b
.
q
co n
mpl t qu
e e
v
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t ien t
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c o rres
h
t is
at
N
J
s ta
a
7
hv
o
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v
JM
ts t o
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g
u
Z
J
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i
s
g
e
+ a,
N
J
C le a rin g
a
a rt s , we
p
u t iv
din g t o t h e p a rt ia l q
n
o
p
'
N
p J
N
q J
r"
wh
N
J
2
co
s it i
o
p
a re
u
b e th e
;
.
t ie n t s
o
N
J
f u th
b , , b2, b 3
a n t itie s a ,
q
2
q
le t
u
GE B R A
7
a , , r, ,
Le
13
V+
ro
.
th e
at
1
AL
fq
d, t ird,
rs t , s e c o n
3 58
GH E R
+ a
.
a
n
d
t ie n t b ”
.
h en c e
+ an p
’
o
an
'
’
r,p
as
0
fl
ra c t io n s
an
d
e
u
q
i g
at n
d irrat io n al
rat io n a l a n
e
’
p ;
en ce a
( rd
—
Rs)
=
d
B u t re — p e
A
rt
sa
e S i gn
[
f
m
th e
S in c e t wo
h
t is in
*
v
n
co n
v
e r en
g
h
o
t
—
’
N
99
,
an
,
4
“
s
p
d i) r— 19 s. pit — M M N a
e n c e a a n d r a re
o s it i
p
h
lds fo r
hv
,
n
re c e de
a ll
To p ro ve th a t th e
.
N9
'
.
e s t iga t io n
3 59
pp
’
co
,
th e
v lu
a
co
es o
mpl t
e e
u
o ti
q
en
h
f n gre at e r t
m
t
l
p
9
—
v
p
e in
t
J
v h
h
mu
h
h f
h
m
th t i
b
h 2 l b i
t
th
2
Thu
th t i
h
mb
v
e
t ege rs
1V + a
.
2
r2
1
an
.
d p a rtia l qu o tien ts
e e an
ha
re c u r
.
N a
A ls o r” a n d
I n A rt 3 57 we a e p ro e d t a t r r ; l
s
t
r _ l a re
i
i
i
n t e e rs
s t b e le s s t a n
N
en c e a "
o
s
t
e
,
p
g
J
a e an
a " c an n o t b e
re a t e r t an
a , , a n d t e re o re it c a n n o t
y
g
'
a l es e x c e t 1 , 2 3 ,
t
a t is , th e n u
er of difieren t va lu es of
b
,
p
a
c a n n o t ex ceed a ,
.
v
n
v u
.
Ill
A ga in ,
au
c an n o
t b
ru b ”
h
v lu
en ce
a
e
11
r” c a n n o
es ex c e
c an n o
rm
bn — a
+1
e
a te r
r
e
g
t b
e
ex c eed
t
g
s
a
an
re a e r
2
t
1
,
,
p
t
fl
a
an
a
2a ,
.
ru
s,
,
n
= a + a
n
n + 1,
'
n
s
a
p
.
h
o s
e
nu
f
er o
hv
hu
h f
iti v i
g
hv
an
s r” c an n o
.
t
n
n
a so
a
z
d t
e
t
e re o re
n
a
dizfi ren t
te
e
er
an
;
y
va lu es
R EC
U R R IN G
hu t h mpl
diff
t v lu
T
”
s
e
qu
e te
co
UE D
C O N TIN
o
N
J
t ie n t
FR A C T I O
+ a,
c an n o
r”
m
2a ,
a
t hat is , s o e o n e
e re n
es ;
th ere
fo re a ll s u b s equ en t o n es , u s t recu r
m
ti en ts
ea ch c
mt
l
us
a so
l
N
J
*
360
To p ro ve th a t
.
We
ha
v
e
a
a
a
rn
u
+ an
b
z
a
p
v
o s it i
e
J
a
es
u
o
q
tien t,
an
d
an
h
fp
o
th e p a rtial
en ce
i l qu o tien ts in
a rt a
.
— l
t h e p ro p o s it io n
r
f
,
“
”
rti rn —
1
N —
a,
ro
e e
h
in t e ge r
N — af
c
o re
o r > i
N
J
wh i h p v
er
r
—
n
l n
a n — l + a ,,
s in c e
m
l
t
p
t
e
a
.
l
l
297
.
hv m
t
+ a it
mb
d th e n u
2
tha n 2a l
rec u r, a n
b e grea ter
c e ca n n o t
y
S
.
A ls o 6" is t h e gre a t e s t in t e ge r in
uo
q
co
N
<i
an
—a <
;
'
r
.
36 1
To s h ew th a t th e p erio d b egin s with th e s eco n d p a rtia l
rs t
u o ti en t an d ter
u o ti en t do u b le of th e
i
t
e
s
w
i
t
h
a
a
r
t
i
al
n
a
q
q
p
*
m
.
fi
v
Sin c e , a s we h a e
l
a c e , le t u s s pp o s e t
p
t en
th e (s
u
h
in A rt 3 59 ,
th
n
at t h e
1
co
(
)
s e en
h
d b
an
we
s
h
a
ll p ro
v
e
re c
a
mpl t
.
e e
,
u
mu t
s
rre n c e
u
q
o
t ie n t
re c
u
.
ta ke
rs a t
b
t h at
( b —l
a
We
hv
a
e
r3 —
— a z
8
e
1
3
u
—l
a —
n
l
n
z
l ,
N
—
a —l
J
a
u
2
+ a8
r —
n
l
r — r
n
n
l
z
b8 — 1 7
z
b
(
n
8
—
_ r
r
—
8
n
l
.
—1
8
—1
)
n
—l
,
—l
z e ro
,
o r an
in t e ge r
.
29 8
B u t , b y A rt 3 6 0,
h u ity
Thu
n
an
h f
f
d th
t
r
an
,
d
an
s
l
u
“h
h
mu
mu
Th is p ro o
.
d
fh
o
u
e z e ro
b
mpl t
co
u
re c
t
ra—
l )
h
at i s
is le s s
en ce
.
'
—
l
n
u
q
e e
o
h f
r;
t
so
on
d
an
al
h
b
a so
if t h e ( n + I )
l
t
i
n t
a ls o
t
e
t
e
o
e
s
p
q
o t ie n t
s t a ls o re c r ;
q
u
an
mu t b
e re o re
en c e
G E BR A
e re o re a _
n
l
s a, _ 1 =
H
AL
al — a —
n
l
.
t
GH E R
HI
t ie n t
re c
(
th e
e re o re
u
th e
rs ,
n
— 1
)
th
th
n
co
co
m
mpl t
e e
.
h
lo n g a s n is n o t le s s t a n 2 [A rt
w
i
h
o t ie n t s re c r, b e in n in
t
h
t
e s ec o n d
g
g
q
lds
as
mpl t
.
u
N
J
u
i
t
f
ll
w
t
h
f
t
h
t
h
u
t
t
I
t
q
w i h th
h ll w h w
d p ti l qu i t b ; w
b gi
h t it mi t w i h p ti l qu ti t 2
N
J
mpl t q u ti t wh i h ju p d th
b th
N
N
J
wh t u ; t h M
mpl t qu ti t
d
N
J
s u tiv mpl qu ti t t h f
tw
d
en c e
t
o
t
n a es
o
a
ar
a
e re o re
s
o
ar
seco n
e
ter
a
u
e e
a1
n s
e
co
en
o
t
th e
t
en
,
o
en
o
en
a
s
e
a,
e co
7
S
st
c
es
re c e
o
e e
+ a,
+ a,
en
o
a re
a,
”
h
r
Ag i
en
rs
re c
i
en
a
n
t
r
b
r
s
h
r
1
—
u
p
ro
p
b
r
o s it io n
h
1
s
e re o re
a
l
en c e a
—a
l
0, t
,,
h
at
is
.
h en c e b"
b
=
en
o
.
is
at
,
a
A ls o
ete
e co
ec
e n c e rn =
an
a n , a
,
co n
+
a
bu t N
e
1C
rl
an
e
‘
co
s eco n
e e
n o
a
.
+ a
e
rre n c e
re c
e
a
:
wh i h
2a
c
e s t ab
lis
hs
e
th e
.
h
e p a rti a l qu o tien ts equ i
d
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To s h ew th a t th e
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18
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e
AL
e rgen
f th e
l 3y2 z
ts
eq
+1
u
o
f th e
rec
atio n s
u
rri n
g
.
lu ti
i p i iv i t g
f
h
Ny 1
m y w pl b y t h
f u d w my b i
f w mh d
S u pp
th t
h i
lu t i
i
h y
d I b i g p i iv
i
t
h
h
wh
i
i
v
1
i
i
;
g
y p
(
g
Thu
37 1
h a s b een
in g
o llo
*
.
W
n
o
et
o se
n
te
e
as
n
e
n
e rs o
as
an
x
e
e
e a se
.
a
e rs
t
os
n
ta
O
a
e
,
o
on
o n e so
en
”
2
x
s a so
z
,
f
en
on
e re
,
t an
,
an
s
n
t
e n
os
t
e
N
J
)
t
os
n
te
e
er
.
s
(
Pu t
x +
N
yJ
y JN
x
z
) (
— le
x
h
(
k yN
2x
h
(
2y J N
h
(
v u
Th e a l e s o f
a s c rib in
t
o n th e
g
c a n b e o b t a in e d
:
)
ky N )
fu
d y so
al e s 1 , 2,
x
N
gy
x
an
o
v u
"
h
(
”
h
(
d
n
h
(
v
i
p
m y lu ti
o s it i
a re
as
an
e
n
so
t e ge rs ,
o n s as
d by
l
a se
e
p
an
we
.
mil
Si
x
a rl
Ny
2
if
y
2
1,
an
x =
h,
d if n
x
hu
T
u
is
*
th e
v lu
re s t ric t e d
37 2
m
b eco
ol e
s v
s
e
.
.
a
es o
to th e
u
u
v
h
i
s
a
l
i
s
o
t
on
o f th e
y
is a n y o dd p o s it i e in t e ge r,
Ny
2
2
h
(
v u
B y p t tin g x
’2
r ’2
x
I
1
,
y
ax
’
,
y
wh i h
c
u
q
a t io n
2
f x a n d y are t h e
al e s 1 , 3 ,
’
e
a
we
y
sa
m
th e
ha
v
e
yf u
e a s a lre a d
eq
u
l
at i o n s a
s
a re a
y
d
o
2
she
w
Np
n
n
d, b u t
?
Z
4
a
“
h o w to
3 08
HI
GH E R
AL
GE B R A
.
i
h
v
u
i
f
t
u
s
N
l
l
wh
i
t
y
q
g
p
d d
mi t
b v m ti
hu w ily
f th
i
t
i
i
t
d
b
2
u
fi
t
i
9
fi d t h t th
y y
q
h
b
f u d y u mb f
lu t i
i i t g
Wh
l
i
i
h
i
d
t
t
l
my b b t i d
lu t i
p
l
i
i
f
h
u
u
i
th t
37 6
S u pp
] ,y g
q
l
i
f
t
h
i
h y
u
u
kb
dl
Ny
y
q
1 ; th
Ny
o s it i
e
a
n
on s
so
a
so
on e
en
e
e o
a
2
x
2
n
n
a
=
as
as ex
x
a
e rs
sa
s a
:
e an
,
W
(f
;
N9
u
B y p t t in g
=
i= N
t
37 7
u
q
a re
or
x
h
if h w v
H it
.
e rt o
o
e
e r,
e r
n
’
"
c
2
t
o se
o
er o
x =
,
e n ex
t
on
o
so
t
o n
o
,
n
.
er o
ar
c e
t
e e
at o n
e e
at o n
.
W )
a,
a
t
h
f
h
g ,
d
a s ex
n
on s
u
f
b
e re
e,
re a e r
e
so
N
a
s
o
—
h
h
(f g )
N
l=
2
l
a in e d i n
p
.
h
u
y v
f
N
it h a s b e e n s p p o s e d t a t
is n o t a p e r e c t !
is a p e r e c t s q a re t h e e q a t io n t a k e s t h e
a
s o l ed as
ic
b
e re a dil
o
ll
o
s
y
(
Pu t
=
es
a
f m
wh
h
m
y
h t
wh
S u pp
b
f wh i h b i t h g
t
th
,
2
’
l
e,
g
y
h i v lu f u
u mb f lu i
in g t o h , 75 t
a
Ob t a in an y n
y
a s c rib
*
s
x
m
we
:
an
so
)
2
—
d
e e as
a
en
x
an
t
n
n
en
s
e
o
ee n
a n e
z
s
= =
E a
,
t
na o rs
s
e
et x
an
;
2
a n e
o se
.
2
x
on
*
x
on
x
en o
on e
en
e
at o n
e e
o
on
o
e a
on e o
s n o
t
so
ra
e
n
e
2
2
x
n
d
an
u
f w
i
i
v
i
p
.
t wo
c a re
o s
t
e
n
t e ge rs ,
en
y)
(
n
x
y)
bc
.
v u
hv
m
u
v u
d
wh
fu
—
i
h
al es o f x an d
b
x
n
f
t
e
c
+ y
,
;
y
y o nd
ro
t es e e q a t io n s a re in t e ge rs w e
a e O b t a in e d o n e s o l t io n
a in in
s o l t io n s
a
b
o f t h e e q a t io n ; t h e re
e o b t a in e d b
g
y
y
a c rib in
t
o b a n d c a ll t e ir
s ib le
al es
o
s
p
g
x
=
n
f mh
u
u
s
m
E x a p le
e q al t o 6 0
u
m
Fin
.
z
d
h
v
two p o s iti
x,
y b
e
in tegers th e
h
du
1 , 60;
d th e
v u
al
e s re q
ct o
u d
ire
h
Thu
t
er e q
s
u
th e
atio n s
n
um
o
en
tain
e
y
2
ifferen c e
o
4 , 1 5;
fro
o
f
2;
v
are
16 , 14 ;
or
v u
al
8, 2
es o
.
h
at
is
f fac to rs
6, 1 0;
5, 1 2 ;
th e
+ y = 3 0,
:
60; t
:
d m
gi in g frac tio n al
b ers
x
2
f th e p air
3, 20 ;
are O b
(ll — y
o
f an y
2, 30;
x
th e
e
th e two in tege rs ; t
N o w 60 is th e p ro
an
.
.
L et
u
eq
u
a tio n s
x
+ y = 1 0,
(1)
—
fx
an
y
6
:
dy
.
o se s q
u
are s
is
I
ND
E TE R M I N A TE
C OR
k m
I n li
o f
.
in t e ge rs
e
f h
if t h e le t
a c t o rs
-
2hx y
b
er c an
e
2
2gx
e
re s o
t
in t o t w o
ed
E
GR E E
3 09
.
in p o s it i
o n
v
e
h,
c =
u i
b
m h d pl i d i
f w i g mpl
I f in t h e ge n e ra l e q
37 8
et
in s t e a d o f e plo in g t h e
ro c e e d a s in t h e
o llo
l
Si
er to
p
p
m
.
m
xa
v
l
2fy
so
DD
lu i
ra t io n a l
lin e a r
.
*
E
SE C O N
b t a in t h e
o
a
by
mmb
d
an
e
O F TH E
my
we
an n e r
ax
f
U A TI O N S
EQ
m
pl
e
y
v
So l
.
v
in p o s iti
e
m
4x
f x , we
s o
4x
2
o
a,
ex
o r
a n e
ex a
n
,
o r
n
e
bot
A rt
h
,
.
a re z e ro
,
3 6 7 it is
.
in tegers
e
2x y
E x p res s in g y i n te r
at o n
2
1 2x
5y
11
.
hv
a e
l 2x + 1 1
2x — 5
In
d h
u
mu t b
s
e eq
Th e
at
al
to
c as e s
O f x are O b
wh
t
o r er
e n ce
th e
al
x
h
1,
4
es o
:
ere fo re
x —
3, y = 1 1 ;
x
dm
th e a
5:
s
u
is s ib le
x
so
—
j
e an
d h
en c e
in te ge r ;
h
e n ce
2x
5
;
th e
dm
a
3;
4
is s ib le
v u
al
es
.
we
3;
u
s
.
re e c te
o
l tio n s
b tain th e
= 4, y
x
9;
:
so
x
u
l tio n s
= 1, y
are
3, y = 11 ;
:
6
1 , 2 x — 5:
4
y
t b
u
m
—5
or 4
cc es s io n
= 2,
2x
be
3 , 2, 4 , 1
are
in
al
3,
or 4
c le arly
a
es
es e
2,
or 4
fx
in te ge r
e an
my
m2
d
v u
h v u
k
dt
a
2, + 6
fro
4
tain e
Ta in g t
an
y my b
x
= 4,
=
9
y
y
.
Th e p rin c ip le s a lre a d e x p la in e d e n a b le u s t o dis c o v e r
i
a dra t ic
at
al e s
o f th e
a ria b le s
i
en
l
n ear
o r
fo r
g
q
a re s
n c t io n s o f x a n d
b
r ect s
ro b le
s o f t is
P
e
c
o
e
e
y
q
p
kin d a re s o e t i e s c a lle d D io p h a n tin e Pro b le s b e c a s e t e y
at e
a t ic ia n D io
an t s
e re
rs t in
e s t i a t e d b y t h e G re e k
p
g
ab o
t th e
iddle o f t h e o rt h c e n t r
37 9
*
fu
w
wh
u
.
v u
v
m
m m
fi
v
m
m
pl
v
u
f
mh m
fu
d
uy
1
Fin th e gen eral e x p re s s io n
c t i s t a en fro
a re s c
t at if t e ir p ro
iffe re n c e is a p e rfe c t s q are
E
d
D
xa
e
uh h
en o
h
.
u
te th e in tegers b y
du
.
x an
x
2
x
h
T is
eq
u
atio n
is
s a tis fi e
m=
x
wh
e re
m
an
d
n
a re
v
p o s iti
d
(
x —
y
b y th e
s
=
u
n
in tege rs
.
s
2
2
—
pp o
y
m
(
r
—
se
;
2
.
p p o s itio n
.
y)
=
v
h
wh h
fo r two p o s iti e in te ge rs
ic
o f t e ir s q are s t h e
th e s u
en
s
y)
n
e
?
m
.
k m
h
zz u
dy; t
xy
.
u
m h
u h
h u
s
m(
z
y) ,
u
310
H
HI
m
Fro
n
x
en c e
mth
e se e q
u
y
GH E R
0,
nz
v
d s in c e th e gi
s o l tio n
u
m
m
m
hm
u
=2
m
atio n
—n2
n
are an
hv
u
en o
=
y
,
o
o
d
ge n e o
m
2
z
v
u
s
my t k f
m— m+
mb i g th g
we
h
eq
u
satis fi e
is
a tio n
m(
+ y ; an
Fro
H
m d
mth
an
e re
n
p o s iti
e
u
atio n s
we
e s e eq
en ce
my t
we
a
n
p
wh
an
v
are
Fro
mth
als o
O dd ;
ree
t
in t e ge rs
al
e ir
e o
al
b tain b y
g
n
c an
f
es
b e fo
n
2 2
)
ud
n
m
(
m(m S
co n
If
6 2 42
m9
:
itio n i s
,
n
u
h
h
= 1, t
um
Th e s
s q a re s o f 6 2 ,
.
s atis fi e
s
d
if
8
x
)
’
-
mS
n
:
h
o
v
p o s iti
ery t wo
ree
v
e
e
in te gers in
is a p erfec t
r
.
m( 4
multipli
c atio n
11
2
m 2m
2
r
,
n
m (m
n
m
an
x
d
n
n
h
h
are e it
is greate r t
an
b
hv
h
t
e
y, t
at
er
o
en o r
is ,
bot
h
n
2
n
3
n
“
0;
.
s are
m
d th e n u b e i s are 4 8 2 , 33 6 2,
3 8 4 4 , 6 7 2 4 , 9 604 , w ic are th e
an
:
v
=
m( m
8m
3 3 6 2, y
o f t e s e tak en i n p aii
re s p ec ti e ly
en
r
.
uh h
n
if
s,
u
4
y
,
h
mu
s
l tio n
it is c le ar t at
st b e s c
t at
x
3
ic
so
,
( in
hu
.
q)
r
c ro s s
2
2
t
.
2
wh h d
(
m m q m+
v u
h v u
e
(q
2
2
re ate r ;
e
d let
p p o s itio n
fo r th e ge n eral
x
h
u
ake
en c e
d th e t
s
n
in te ge rs
o
eral
2 q2 ,
?
b y th e
)
I
wh
d
=n
a
'
t
mf
— 92
1
T is
th e ge n
u
e n
2
2
=
=
=
2
2
x
x
+
,
,
q
y p
y
p
or
e
= 33,
x, x
y,
a
n
z
uh h
x
a
in te ge rs ,
e
y
2x —
.
m
y two p o s iti
te th e in tegers b y
0
:
catio n
hm
is
are .
D
2
multipli
c ro s s
Fin th e ge n e ral e x p re s s io n fo
E x a p le 2
t at t h e s u
e tic p ro gre s s io n , an d s c
arit
sq
m
)y
n
z
a e
.
.
n
n
n
eq
x
H ere
an d n
= 7 , n = 4 , we
m
en
(m
y
x
an
G E BR A
nx
we Ob tain b y
atio n s
2
AL
h h
XX I X
C H A PTE R
U M M A TI O N
S
.
OF SE R I E S
.
u mm i f i i h v u d
i
l
b
i
l
v
u
h
i
w
v
t
h
t
i
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t
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i
i
;
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p
p
y p i f th m h d f u mmti wh i h h v l dy b
i
d
l
p
i
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V
C
h
h
i
l
P
m
i
t
i
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p
g
()
i
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h
P
l
i
m
t
G
ii
p
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( )
i
m
l
i
l
d
l
wh
h
y
i h
i
i
ii
S
y
p
p
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( )
mt i l A 6 0
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68 t 7 5
A t
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312
A
V
i
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I
u
h
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i
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p
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( )
w p
mh d fg t g
d t di u
lit y
W
u f th p
h p it w ill b
h m
h
i
b
ill
d
m
b
ll
i
m
h
u
fu
y
m
l
y
d
f h f
y
g
g
p
E xa
3 80
.
a n e
ex
at o n
et
s
o
s
o
t
e
en
en
on
a
rt
,
r s
er es,
re s s o n
ro
e r ca
a
e re
c
a
e
e o cc
rre
o
e
a
een
a re a
a
,
ar
.
.
.
.
t
et c a
a rt
an
eo
.
.
o
.
t
e rs o
o
e
at
e
ra
e rs a n
a
e
.
at o n
rt .
a
,
a rt
a re
c
s O
v
res s o n
ro
ca
er es
ean s
e er
n
o
n e
oe
c en
ts,
.
t
n
t
e
ro c e e
n o
e
t
rs e o
e co
o re
er e s,
rr n
ec
v
o
co n
se r e s
.
eo
u
c e rt a n
o
e
e rs
a
e
r
e r ca
fs
es o
c
o
s s
n o
s
s
o
re
n
mpl
o
et
n
a
o
sc
ss
e
re s e n
t
s
o
a
et
st
o
s
o
re a e r
te r
a
c
e
m
th
.
.
e n e ra
e s een
se
e
o
e
t
at s o
e
.
o f a s e rie s c an b e e x re s s e d a s t h e dif
I f t h e r te r
p
ic is t h e s a e
n c t io n o f r
e re n c e o f t wo q a n t it ie s o n e o f
o f t h e s e rie s
l , th e su
a
b
re a dil
e
t a t t h e o t e r is o f r
y
f
h
fu
o
381
n
.
u
h
d
m
s e rie s
b e de n o t e d b y
u
d it s
th e
y
.
Fo r le t t h e
an
m fu
m
wh h
su
f m
v,
o r
S
=
n
mb y
(
S
—v
t
h
d
s
— v
0
.
2
u pp
+ u
3
o se
t
h
at
an
m
r
t
e
y
can
a
r
b e p u t in
en
(
v,
v
an
l
+ u
v —v
s
2
)
+
_ v
S
m
E x a p le
mt
Su
.
U M M A TI O N
m th
ter
o n
s
d
I f we
en o
1
1
2
x
+
)
(
1
2
x)
+
(
te th e
s e rie s
1
( 1 + 3x)
by
u
we
313
.
e s e rie s
1
1
( + x)
O F SE R I E S
1
+u2+
u
3
+
hv
a e
1
1
1
1+
x
by
a
dd
itio n
,
+ 1
n
.
x
1
l
1
x
l +x
1 +n + 1
.
x
n
m m
u
my b
mt h d
f m
So e t i e s a s it ab le t ra n s o r a t io n
b y s e pa ra t in g u in t o p a rt ia l ra c t io n s b y t h e
in C a p X X I I I
382
h
E
xa
.
.
e
Fin
.
d
th e
Th e
n
)
‘h
l
( +
ax
a
)
x
m
a
ter
B y p t tin g 1
a
"“
x,
n
‘
n
ml
i arly ,
to
n
te r
Ob
ta in
l
(1
1 + a "x
I
“2
2
)
eq
rt
n
a x
u
al
)
to
z ero
’
—
l
a
(
a
in
a
s
'
a
a
a
1
ax
1
n
u
n
l —a
m
1
(1
B
—1
1 —a
Si
s
m
l
-
a
H en ce
o
o
a
a
u
b t a in e d
e x la in e d
p
o
mf
su
1
x
e
e
.
m
pl
(l +
f
a
2
2
a x
‘
l
x
)
.
cce s s io n
,
we
s
.
31 4
HI
GH E R
m
AL
G EB R A
.
m
m
m
m
n ter
o
a
r
i
e
a
h
o
To fin d th e s u
s
s
e
s
e
h
t
r
o
h
i
c
c
e
w
f
f
f
i s c o p o s e d of r fa c to rs i n a ri th etica l p ro gres s i o n , th e rs t fac to rs
e a ri th
etic a l
ro gres s io n
h
evera l ter
s b ein g i n th e s a
o
t
e
s
p
f
83
m
wh
.
m
b e de n o t e d b y
s e rie s
L et t h e
m
u 1
u 2
e re
_
R e p la c in g
(
=
u
by
n
b)
by
e re o re ,
(
a + n
b)
(
—
s
ub
1 we
n
t ra c t io n
hv
a
mil
By
h
at
(
ly ,
ar
a
ddit io n ,
is ,
(
“ 11
-
1
(
1) b
u 2
(
r
1) b
u 1
(
r
1) b
Sn
—
+1
G is
a s c rib
a
in g t o
Th e
a
Write
bo
n
v
u
q
so
an
m
e
e re s
9
r +
yi
n
u l giv
t
u
sa
y ;
v
sa
y
b) ;
.
m
i
i t,
2
“
u
"
G,
1) b
de p e n de n t o f n
al e
a rt ic la r
p
t it
b)
—2
1) b
r +
(
e re
.
+ r
v1
b)
wh
1
a + n
4
r
”n
.
,
1) b
r
.
e
“ 7. 2
Si
(
I b)
a + n + r—
e
a
a + n
R e p lac in g n b y
h f
hv
1 , we
u
1
a + n —
(
t
.
u 3
b)
t
fi
u
es u s
wh i h my b f
c
,
v u
t h f ll w i
e
a
e
o
o
n
g
co n
v
e n ie n
m
m
d by
.
t
do wn th e n ter , afi x th e n ex t fa c to r a t th e
b y th e n u b er qffa c to rs th u s i n crea s ed a n d b y th e c o
en c e, a n d a dd a c o n s ta n t
th
o un
ul
r
en
e
d, d ivide
o n dif
xr
mm
.
It
hwv
my
a
b
e
b e tt e r
in dic a t e d
o
e
er
.
n o
n o
t ic e d t
t to q
u
o
h
at
v,
G
h
t e t is
(
re s
r
ul
a
(
l) b
t, b
u
t to
o
r
l) 6
b t a in 0
Q6
it
as
ab o
is
v
e
316
GH E R
HI
AL
m
G EBR A
.
m
m
m
of wh i c h
To fi n d th e s u
of n ter
s of a s eri es e a ch ter
is co p o s ed of th e recip ro c a l of th e p ro du c t of r fa c to rs i n a ri th
s b ein
n
i
e ti ca l
rs t fa c to rs qf th e s evera l ter
ro gress io n , th e
g
p
the sa e a ri th e ti c a l p ro gres s io n
3 86
m
m
wh
.
m
fi
m
.
s e rie s
L et t h e
b
de n o te d b y
e
u l
u
as
.
e re
1
:
(
R e pla c in g
by
n
a + n
+ r
—
3
a + n
R e pla c in g
b)
1
+ r
by
n
h f
by
e re o re ,
ub t
ra c t io n
(
mil
Si
s
y
(
a rl
.
hv
a
a
+n
(
r
(
r
(
e
u
u
n
n
1) b
r
>
b s
-
,
1) b
r
b)
2
b)
1
l , we
n
(
t
b)
1,
n
1
“
—l
vn
4
_1
r
By
h ti
wh
t
ditio n ,
ad
s
a
e re
G is
a s c rib in
hu
a
t
o
g
T
s
H
en c e
n
u
q
so
an
e
vl
— v
n +1
(
—1 b
)
r
yi
vl
z
(
de p e n de n t
al e
a rt ic la r
p
t it
m
Sn
1) b
n
u
v
u
o
1) b
r —
fn
,
wh i h
f
c
o u n
d
.
G
S ,,
th e
su
Wri te
mmy
a
be
f
o
u
n
d b y th e
m
f
o
llo win g
ru
le
m
do wn th e n ter , s tri ke of a fa cto r fro
th e b egi n
divide b y th e n u b er offa cto rs s o di in is h ed a n d b y th e co
dif eren c e, cha n ge th e sign a n d add a c o n s ta n t
m
th
m
n
i n g,
mm
on
.
Th e
v
e ac h c as e
u
al
e o
a
fG
mi
t o de t e r
n e
0 by
a s c rib
T
b
in g t o
n
so
mp
e
a rt ic u
la r
v lu
a
e
.
S
m
E x a p le 1
d
Fin
.
U M M A TI ON
th e
su
mf
o
Th e
h
n
th
n
s e rie s
1
1
r
le , we
hv
a
Sa
Pu t
f th e
317
.
lb
u
b y th e
e n ce ,
s o
1
m
te l
m
te r
n
1
OF SE R I E S
= 1, t
h
e
C
z
1
_
3
(n
+ 1)
(
+ 2)
n
+ 3)
(n
wh
1
en
C
en ce
1
1
“
18
mki
By
d
g n in
n
a
m
E x a p le 2
efi n
Fin
.
d
itely great, we
th e
su
mt
o
u
d
n
n
(
u
n
(
n
(
n
h
wh h
+ 3)
(
n
(
n
(
n
+ 3)
-
3
+ 1)
(
my
hu
a
4
+ 2)
n
n o
(n
+ 3)
n
k
w b e ta
en
(n
as
+ 1
)
th e
n
mf
th
ter
O
.
3
4
—
n
h
+ 2)
+ 1 ) + 3n + 4
f t e se exp re s s io n s
th e r le is applic ab le
t
m
+ 2)
T
1
huh
hm
h m
.
+ 1)
n
1
=
u
t
n
p
s erie s
u
m
(n
(u
o
f th e
5
1
E ac
ic
s o
18
ire c tly app lic ab le , b e c a s e alt o g 1 , 2 , 3 ,
in a t o rs , a re in a rit
e t ic al p ro gre s s io n ,
e ral
en o
in ato r are n o t
I n t is e x a p le we
en o
ay
+2
+ 1)
m
t er
n
d
v d
d m
w
b tain S c ,
4
3
H e re th e r le is n o t
th e fi rs t fac to rs o f th e s e
th e fac to rs o f a n y o n e
p ro cee a s fo llo s
o
1
+3
2
(n
+ 2)
(
n
+ 3)
3
(
n
+ 1)
+ 2)
(n
(
n
+ 3)
en
1
3
4
3
Wh en
4
1
29
36
n
+3
ce
4
3
2
c
3
(u
+ 1)
(n
+ 2)
(n
+ 3)
a s erie s
31 8
HI
GH E R
mh
h
GE B R A
AL
.
e t o ds o f A rt s 3 8 3 , 3 8 6 a re direc tly
I n c a s e s w e re t h e
387
a
a lwa s e fe c t t h e
l
i
c a b le , in s t e a d o f q o t in g t h e r le s w e
a
y
pp
‘
ic is s o e t i e s c alle d t h e
t h e o llo in g w a y ,
a t io n in
s
’
M e t o d o f S b t ra c t io n
u
f w
.
u mm
h
E
xa
u
m
pl
.
d
Fin
e.
m
u
y f
m m
wh h
.
th e
su
mf
o
m
te r
n
f th e
s o
14 +
2
hm
e tic a l
arit
Th e
s e rie s
p ro
h
gres s io n in t is
is
c ase
2 , 5, 8 , 1 1 ,
h m
hm
v
du
o f th e gi e n s e rie s in tro
te r
I n e ac
en o te t is
e tic al p ro gre s s io n ;
o f th e arit
b y S ; t en
h
S
S
’
2
z
d
h
5
.
s
u
b trac tio n
5
—2
.
S=
yt
re a dil
F
di
m
h
W
o
.
a
o
or s
f
it
n
y
l
pp
u pp
c an
th e
)
.
d
5] — ( 3 n
.
5 9,
.
m
f a s e rie s is a ra t io n a l in t e gra l
ic
ill e n a b le u s
b e e x p re s s e d in a o r
i
3
8
3
e n in A rt
et ho d
g
n
te r
o
m
f mwh h w
v
is
n
ass
(3 n
s
3
a + 5)
(
th
( )
<
1)
o se
e n s i o n s , an
s
n
th e
en
m]
—1 t e r
9 S = (3 n
.
m
— l te r
)
n
,
2
388
c t io n
v
+ ( 3n
(
14+
n
s erie s
14 +
.
m
fac to r th e n e xt t e r
’
b y S , an d th e gi en s e rie s
'
By
fu
c e as a n e w
ra t io n a l
a
um
.
.
in t e gra l
fu
n c t io n
o
f
o
n
fp
e
wh A B C D
u d mi d
i
1
p
u mb
u f ll v lu f w my qu
Th i id t i y b i g
f lik p w
f ; w th u
ffi i
b
i p
h
l
1 im
p
i
h
m
u
i
t
d
1
p
q
m
m
wh
m F d
m
A um
e re
,
,
,
a re
,
c o n s ta n
n e
ts
+
n
er
n
.
t
e co e
c en
at o n s
e
E
p le
xa
t
en
s
ts
o
in
.
t
n e
th e
su
ss
it is
u
o r a
e
o
e
et er
o
tr
e n
e rs O
o
es
a
c o n s ta n
f
n
te r
s
4
o
o
n
ts
f th e
e
,
ta
s o
e
n
e
n
s
e te r
n
e
ate
S
e
a
+
n
.
s e rie s
o se
ge n
e ral
te r
is
5n
e
at o n c e o b
v u h
io s t at A
0, E
0, E
cc e s s i e ly , we o b tain C
6, D = 0
T
v
:
:
:
n
4
+ 6n
3
+ 5n
.
2
n
(u
+ 1)
(
n
1;
hu
s
+ 2)
(
n
an
u
d b y p ttin g i t
+ 3)
6n
2,
n =
3
3 20
HI
GH E R
AL
G EBR A
.
m
m
m
"
11
th e r
of n ter
s of
r
a n d th e s u
d
t
h
e
n
t
e
fi
b e rs
o rder qffi gu ra te n u
h
'
“
th
l ; th e n
ter
o f th e
rs t o rde r is
te r
o f th e
Th e n
th
o f t h e t ird o rde r is En , t a t is
ter
s e c o n d o rde r is n ; t h e n
n
1
n
)
(
th
rt
o rde r is 2
t a t is
o
h
t
er
o f th e
t
e n
n
n
(
392
To
.
n
m
th
m
fi
m
m
(
n
+
u
1)
(
2)
n +
t
,
t
h
h
ter
2
h
fu h
m f th
th
e n
o
an
hu
s
it is
eas
y
to
t
see
h
th e
at
d
(
E
n
'
n
so
o n
te r
o
th
e r
n + r
)
A ga in
,
th e
su
mf
o
n
wh i h i
c
N O TE
an
d
s
.
o r er
y
Z CI O
th e
n
th
r
(
1)
(
n +
2)
o rde r
is
—2
— l
m
ter
n
u +
1)
(
m f th
ter
n +
.
m f th
th
h
h
i2
e i i ft ii o rde r i S
‘
7
at
T
m
.
o
s o
n +
e
u
(
th
f th e
2)
(
1)
r
r
th
n
o rde r
+ r—
is
1)
o rde r
.
m
m
o f n te r
I n ap p ly in g th e r le o f A rt 3 83 t o fi n d th e s u
b ers , it ill b e fo n
t at th e c o n s tan t is al
o f fi gu ra te n
um
w
.
ud h
s o
w
f
ay s
’
.
f
u ra te
u
n
fi
g
p
a de
in t e re s t in g o n a c c o n t o f t h e u s e
h is Tra i té da tri a n gle a ri th é tiqu e, p b lis
39 3
Th e
l
st
e
p
m
si
Th e
.
ro
f ll w i
f m
o
o
o r
n
rt ie s
e
p
u
a b le
t
g
o
m
m
ex
h ib it
mb
u h
s
t h e A ri th
ers
hi t
y
s o ric a ll
a re
hm
f t e
b y Pas c a l in
e d in 1 6 6 5
o
.
mti
e
ca l
Tria n gle in it s
S
u
Pa s c a l
c o n s tr
ul
in g
r
th
u
n
u mb
si
er
32 1
.
f ll w
t h e t ria n gle b y t h e
n
o
o
m
mf
th a t
o
i
mmdi
e
a tely a b o ve
it
an
d th a t
s
F mt h mde
iv h i
u
s
u mb
c cess
o
e
u
f
um
s
f c o n s t r c t io n , it o llo w t h a t t h e n
b e rs in
o r z o n t a l ro w , o r e rtic a l c o l
n s , a re t h e fi u ra t e
g
rs t , s e c o n d, t ird,
o rde rs
e
o
s
fi
w
f th e
e rs o
n
th e
E a ch n u b er is th e s u
edia tely to th e left o
i
t
f ;
ro
th e
c t ed
O F SE R I E S
e
mm
i
U M MA TI ON
v
h
um
.
um
u
f m
u
A lin e dra n s o a s t o c u t O ff a n e q al n
b e r o f n it s ro
t h e t o p ro w a n d t h e le t h an d c o l
n
is c a lle d a b a s e, a n d t h e
b a se s a re n
b e re d b egin n in g ro
t h e t o p le t— a n d c o rn e r
Th s t h e 6 t h b a s e is a lin e dra wn t h ro gh t h e n
b e rs 1 , 5, 1 0,
1 0, 5, 1 an d it will b e o b er e d t h at t e re a re Six o f t h e s e n u
b e r , a n d t h at t e a re t h e c o ef c ien t o f t h e te r
in t h e e x
a n s io n o f 1
p
(
f
um
u
s
um
f m
-
s v
hy
s
h
f h
um
ms
u
h
s
fi
um
w
s
y
.
m
u
Th e p ro p e rt ie o f t e se n
e re dis c s s e d b y Pa s c a l
b e rs
it gre a t s k ill : in pa rtic la r h e s e d h is A rith eti c a l Tria n gle
t o de e lo p t h e t h e o r
so
b in at io n , a n d t o e s t a b lis
e
o f Co
in t e re s tin g p ro p o s it io n s in Pro b ab ilit
ll
Th e s b e c t is
’
t re ate d in To d
n ter
H is to ry of Pro b a b i li ty , C h apte r I I
w h
v
u
y
s
hu
u
m
m
h m
fu y
uj
.
.
s t th u mb f t ms
y mb l 2 t i di t u mm i ;
i s w h v u d h
i
h f ll w i g mdifi d
t ti
w h i h i di t
m
b t i
w hi h th u mmti i t b ff t d w ill b
i
b tw
t h lim
i t
f u dm
m
i
f
t
h
m
f
u
t
t
h
d
s
t
L t l( ) b
2
4( )
y
ll p i
i
f t m b t i d f mI ( ) b y giv i g t
f th
f ml t mi lu iv
t iv i t g l v lu
u pp i i qu i d t fi d th m f ll th
F i t
s f th i s b t i d f mth p i
t m
39 4
n
a
u
ser e
e
,
ts
a
n o
t
mb igu ity
e
o
ee n
e
e x is t
s
e
as
o
o
e
n
n o
a
ca e
on
er
at o n
s
c
,
e e
o
s
on
a
e s
er o
e n
o
o
n
o
c
o re c o n v e n e n
n
t
se
e
a
e c a s es
so
n
e
o
W h ere
.
c a es
n
ec e
e
,
.
xz
e
e
x
c)
e
n
o r
er
es
a
ra
e
s O
er
e s e r es o
O
e ser e
v
b y gi in g t o p
H
.
H
.
A
.
—1
l in t e gral
al
t
en
ro
n c
s re
)
( )
x
s
e
re
) (p
-
(r
v lu f m
a
es
ro
r
x
en o
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e
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ro
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—r
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s
e
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e
3 22
HI
f
Writ in g t h e
th e
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1
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um
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i
S in c e t h e gi
in c l s i e , we
p
v
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l ) ( v— 2 )
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my w
mth
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is
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E XA M PLE S
XXI X
.
m
e
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ter
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rt 3 8 3
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r
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o
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we
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S o fa r a s we a e p ro c ee de d, t h e n
e ric a l c o e f c ie n t s
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t h e s a e la w a s t h o s e o f t h e Bin o ial t eo re
W e s all n o w
ro e b
i
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a t t is
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l
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h
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g
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we
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m
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o
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in
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if t h e la w o f o r a t io n
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e re
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rz
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)
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u 2
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f di fe ren c es
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ide n tic a ll 3
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s
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t h e la w
o
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on
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t
h
at
sa
m
n
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1
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is ,
+
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e as
in t h e p re c e din g a rt ic le ;
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1
l
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2
;
n
n
n u1
A u 2,
+
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m
v
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l
-
I
m
lae o f t is a n d t h e p re c e din g a rt ic le
a
b
e ex
Th e o r
y
i
f
rs t
d
e re n t
s : if a is t h e
re s s e d in a s li h tl
o r , as
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g
p
s o f th e su c
rs t t e r
th e
ter
o f a gi e n s e rie s , dl , d2 , d3
th
fe re n c e s , t h e n
c e s s iv e o rde rs o f dif
ter
o f t h e gi e n s e rie s is
la
th e o r
o b t a in e d ro
f m f w
fi
m
m
v
fi
3 26
an
H I
d the
GH E R
G E BR A
AL
.
m fn t mi
su
o
(
u
n
s
s
er
— l
)(
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n
(
n
u —
l ) (n — 2 ) ( u
3)
—
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E
xa
Th e
l
m
p
s
u
d
Fin
e.
m
th e gen eral ter
v d
c c e s s i e o r ers o
an
d
f i ffere n ce
d th e
su
m fn t m
er
o
f th e
s o
s e rie s
are
2 8 , 50, 7 8 , 1 1 2 , 1 52 ,
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e n ce
th e
n
m 12
th
ter
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tain
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.
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th e fo r
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my
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s
o f n ter
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3
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2 71 + 52 n + 6 2 7 u
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v u
g o n th e al
o f th e p res en t artic le
+ 1)
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y
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f m
v u y m
h h
m
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m
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w
i
l
l
i
l
i
y
w
d
fu
i
f
h
d
i
m
im
i
t
i
F
p
;
f h w v i p f ly g l
h mt h d f p
w
h
et o d o f s
a t io n
I t ill b e s e en t a t t is
ill o n l
397
or
in g t h e o rde rs o f
t a t in
s c c eed w e n
t h e s e rie s is s c
diffe re n c e s w e e e n t all c o e t o a s e rie s in w ic a ll t h e t e r s
th
il
al
a s b e t h e c a s e if t h e n
i
l
t
er
al
T
s
o f th e
a re e
q
n c t io n o f u
s e rie s is a ra t io n a l in t e gra l
.
.
.
or S
t
e
e
c
L e t th e
u
wh
l
s e rie s
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e
ro o
o
o
t
2
+ u
o
,
z
e
e r,
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+
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n
s
n c
e r ect
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on
t
o
e n e ra
re e
en s o n s
.
n
th
0
+1
a
s
de n o t e t h e
d le t 1) w",
fe re n c e s ;
o rde rs o f dif
an
er a
be
u
e re
co n s
Cu
m
ter
o
0
0
0
0
0
D,
f th e
fi
rs t,
se c o n
h
d, t ird
328
H
1 —
n e ral
e
g
mi
t er
fi
s
c o e f c ien
ts
at
t
s
o
o
fi c ie n
c o ef
th e
f th e
o
f ll w h
it
en c e
x,
a re
GH E R
HI
ts
o
f
x
fi
th e
n
a
"
f
AL
GE BR A
.
u iv mu ltipli ti s b y
d thi d
fi t
u
d
p
d t hi d
d
f diff
t e r th e
in t h e
s
c c e ss
rs
rs t , s e c o n
,
s ec o n
r
,
e
ca
r
,
o r e rs o
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.
es
o
e a
a rr
er
rt 3 9 7
A
[
hu
wh
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ter
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t
h
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,
re s s o n
n
c o n s t an
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n c
on
o
h
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m
.
)
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s t an
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x
"
m
at
re la t io n
is
ds fo r t h e p t e r
e n d o f t h e p ro d c t
th e
at
e
f (x )
d
an
d p ter
an
I
u
s
.
)
l
(
t
at i s ,
P“
—
1
x
)
(
is
s e rie s
th e
s
er
an
1w
S<
t
o
s
er
ra
S (l
s
h is
e re
a
e
n
]
.
T
n
n n
e
e
e r ca
eo
a
f n ofp
c at o n s
x w e S a ll
c ,
t e e x c ep t io n o f p
at
e en
o
t e s e rie s , o r
o se c o e
c e n s is t h e s a
e
on a
ra
s
e
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a
s a
a
an
ere o re
en s o n s
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ro
,
l i t g l fu t i
ti
hyp t h is i
l
i
li
i
m
t
b
h
f
f
u
1
t
dim i
p
p
;
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i s t h t m f wh i h w it h h
iv
t
th
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dp t m
d f h
t m t t h b gi
f
f
i
i
h
wh
fi
t
m
i
l
t
p g
g
By
on
a
re c
A
rt 32 5
[
]
1
(
u
rrin
wh
s e rie s
g
o se
s c a le
o
f
.
m
mh
m
v
is n o t gi e n , t h e di e n s io n
I f t h e ge n e ra l t e r
t
h
re a dil
o n d b
e
e t o d e x p la in e d in A rt 3 9 7
y
yf
E
xa
u
m
pl
e
Fin
.
d
u
th e gen eratin g f
3
m
Fo r in g th e
s
th e s erie s
u
.
n c tio n
2
1 5x 3
v d
f
5x +
9x
cc e s s i e o r ers o
o
f th e
s
o
f
a
a re
n
.
s erie s
2 3x 4 + 3 3 x 5 +
d
ifferen ces
o
f th e
c o e ffi c ien
ts , we
hv
a
e
is
a
2 , 4 , 6 , 8 , 1 0,
hu
m
d d
hv
th e ter s in th e s ec o n
o r er o f
ratio n al in te gral f n c tio n o f n o f two
3
1
o f relatio n i s (
x
We a e
)
t
s
u
.
d
dm
u h
h
iffere n ce s are e q al ,
en c e
i e n s io n s ; an d t ere fo re t
S = 3 + 5x +
3x 8
9x
1 5x
2
27x
3
2
3x 3
"
By
a
dd
itio n
5
3
,
3
3
3x
“
(1
3
-
4
4 5x
27 x
4
5
4 542
5x
4
5
9x
— 4 x + 3x 2 ;
2
—
3 4x + 3x
—x 3
1
)
(
an
h
e s c ale
S
U MM A TI O N
hv
u
o r SE R IE S
h
3 29
.
h
f
h h
W e a e s ee n in C a p XX I V t a t t h e ge n e ra tin g
39 9
o s e de n o
n c t io n o f a re c rrin g s e rie s is a ra t io n a l ra c t io n
i
S pp o s e t a t t is de n o in a t o r c a n
n a t o r is t h e s c a le o f re la t io n
bx) ( l
ex
b e re s o l e d in t o t h e a c t o rs ( 1 a x ) ( l
t
e
n th e
)
a rt ia l rac t io n s o f t h e
n c t io n c a n b e s e a ra t e d in t o
n e ra t in
e
p
p
g
g
.
fu
v
fu
f m
E h f th
f
h
i
f mf
u ig i
ac
o
t
e
n
es e
or
l
o
eo
s
h
cx
m
m
h
b e e x p an de d b y t h e B in o ia l T e o re
e n c e in t is c a s e t h e re
e t ric a l s e ries ;
of a n
b er o f
e x p re s e d a s t h e s u
can
m
g
c rr n
s e r e s c an
be
eo
e t ric a l s e rie s
g
m
l —
l — bx
— dx
a
m
0
B
ra c t io n
m
wh
f
A
or
.
u
.
f
.
h
s
h
m
um
.
w
f
—
a
n
i
i
n
s
a
n
l
a
t
o
c
o
n
t
a
c
t
r
f
r
e
o
l
d
x
o
y
o re t a n o n c e , c o rre s o n din
ill
o t is re e a t e d a c t o r t e re
b
e
t
p
p
g
A3
A3
a rt ia l
rac t io n s
i
c
o
f
t
h
e
o
r
3
p
l
ax
1
ax
)
(
(
)
w e n e x pa n de d b y t h e B in o ia l T e o re
do n o t o r geo e t ric a l
s e rie s ;
e n c e in t is c a e t h e re c rrin
s e rie s c a n n o t b e e x p res s e d
g
as th e s u
ofa n
b e r o f ge o e t ric a l s e rie s
m
If ho
h
th e
e ve r
s c ale
f m
f
h
h
4 00
.
p
ro
m
h s
m
um
Th
iv
u
e s
c c es s
u
f
m
e o rde rs o
(
l ),
r
a
wh i h
c
a re
mm
on
4 01
.
f
f di fe re n c e s
.
t
(
r
h m lv
e
ra t i o
Le t
se
r as
us
a r,
a
(
ar
— I
th e
ar
,
l ) r,
r
es
3
g
f
,
a
(
a
(
r
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4
ar
o
f t h e ge o
mt
e ric a l
( 67
l)r
r
)
2
r,
2
n
i
5
q( ) s
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t is
E ac te r
in
a
,
a
(
(
s
l) r
r
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1)
m
2
r
ge o et ric al p ro gre s s io n s
o ri in a l s e rie s
g
,
2
,
h vi
a
n
sa
g
m
e
.
c o n s ide r
th e
s e ries
in
wh i h
n
5
9 (
h
w
wh h
g
a
w
h
f m m
m
h
re s s io n
a,
co
h
a ra t io n a l
c
)
,
m
fu
in t e gra l
n c t io n o f n
e n s io n s ,
o f p di
s e rie s le t u s o r
t h e s c c e ss i e o rde rs o f diffe re n c e s
an
o f t e s e o rde rs is t h e s u
a rts , o n e
o f t wo
y
p
a ris in
ro
t
r
e
o
f
h
t
e
r
o
an d t h e o t e r ro
te r s o f
g
th e o r
N o w s in c e g6 ( n ) is o f 79
¢ ( n ) in t h e o rigin al s e rie
di e n s io n s , t h e p a rt a ris in g ro
ill
n
b
e z e ro in t h e p
(
(
)
an d s cc e e din
o rde rs o f dif
fe re n c e s , an d t e re o re t e s e s e rie s
g
ill b e ge o e t ric a l p ro gre s s io n s
h o se c o
o n ra t io is r
A
rt
[
e re
f
h
f
m
w
u
mh
m
f m ms
m
m
f m u
v
h
m
f m
h f m m
s
f mM w
h f h
w
mm
.
.
.
.
3 30
GH ER
HI
G
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H I
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AL
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m
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t
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en
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XXIX
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x7
5
5
1
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fo llo win g s eries
e
o
5
x
3
x
x
m f th
su
G EB R A
su
o
s
.
.
HI G HER
340
24
I f A , is th e
.
AL
GE BR A
"
c o effi c ie n t o f x in th e
.
e x p an s io n
f
o
0 0 0 0 0
t
r
o
e
v
p
h
at
2
I f u is
25
.
n
(
a
an
T
multipl
e o
f6,
l ) ( n —2 )
u —
n
(
s
h w th
e
u —
n
is
eq
al
to
z ero
If
ni
.
p
is e qu al to
n
n —
2)
1
n
4)
+
3
(
l ) ( n — 2 ) (u — 3 ) ( u — 4 )
u —
1
0
lg
i
i
o
s
t
v
e
p
in teger,
s
h w th
e
n
_
1
P
“
n
3
2
ooo o
q
)
at
n —
é
3)
PQ
+1
?
—r
)(
—r
+2
n
(H
s
—
s erie s
L
If
.
f th e
l ) (n — 2 ) ( n — 3 ) (u
— 2
(
o
.
s a
—
27
)(
h
3 15
5
lg
u
26
(
—l
d A4
at eac
E
u
1 07 2
)
(
n — r+
p
-
l),
g
h ew th at
P1 Q 1 + P2 Q 2 + P3 Q 3 +
28
I f n is
.
77. —
u —
(_
3
a
u
29
ac c o r
a
p ro p er
3
x
— x6
-
1Q
2
n —1
7
h w th t
—5
)(
e
a
—6
n
)(
u
—
7)
e
7 2’
1
u
di g
— or
x
1 — x2
(
3
If x is
.
e o f 3, s
H
2
is eq al t o
multipl
4 ) ( u — 5)
‘
+ Pn
—1+ +
p
q
9 L
LL
n
0
f
as u
rac tio n
x5
l —
n
x
m
,
s
is
o dd o r e
hwh
e
t
v
en
.
at
x
1 + x2
+
x
3
1+x
x5
6
?
o
o
o
C H A PTE R
T
v
4 07
l
t in
a en
h
h
UM B E R S
N
OF
Y
.
h
.
w
I n t is c a p te r we s a ll u s e t h e
e a n in g t o p o s iti ve in teger
m
u mb
.
HE O R
XXX
mb
o rd n u
er a s e
u
i
q
.
ly div i ib l b y
wh i h i t
u
m
b
y
i
t
i
ll
d
i
m
t
i
t
l
f
d
u
y
m
m
b
i
p
p
p
u mb w h i h i divi ib l b y h u mb b id i lf d
m
m
u ity i ll d
i
b ; t hu 53 i
i
m
u
m
b
p
p
mp it u mb
Tw
u mb wh i h h v
d 35 i
i
i
mm f t
t
d
t
b
i
m
u
y
h
h
t
p
;
p
t hu 2 4 i p im t 7 7
f t h f ll w i g l m t y
W h ll mk f qu t
4 08
m
i
i
ll
i
h
i
f
wh
t
u
y
t
f
h
d
fi
i
t
i
t
t
p p
i
m
f p im t h t t h y my b
d
d
g
i
i
I
f
m
b
d
i
i
m
u
v
s
u
d
d
i
O
t
d
t
b
p
p
()
f t b it mu t divid t h th f t
F
i
div id
v yf t f i f u di b ; b t
b
i
i p im t b
f t f i f u d i 6 ; t h f ll
h f t
f
f u d i ; t h t i divid
ii
f
i
i
i
m
I
u
v
d
u
u
i
m
m
b
d
d
p
( )
p
f t s f t h t p du t ; d t h f if
div id
f th
i
i
i
i
i
m
m
i
i
t
t
u
v
w
v
d
id
h
b
b
p
g
y p
mu t divid 6
i
i
m
i
t
m
b
d
i
i
i
h
f
t
h
u
I
f
i
i
m
6
p
p
( )
div id b
h p du t b
f t f
h
t
F
f
f h p du t b i t div i ib l b y y f
hti
v ly if i p im t b i i p im
h
i p im t b
C
f h
um
b
b
d
Al
if
h f th u mb b l it i
i p im t
i
m
m
t
i
i
l
i
u
v
y
f
t
h
t
d
t
d
b
d
y
p
p
p
b
f th t um
u mb it i p im v y f
A
er
n
ex ce
er
n
n
ca
an
s
s
s
e s
.
o
os
a
e
e
e
a c o rs O
r
e
e
o re
r
s
t
o
t
e
o
e n
so
r
n
e
o
e r,
n o
ac
n
n
ar
e
es
r
er
e
ts e
es
an
e r,
c
o
a
;
e n
r
e
er
c
a
eac
n o
e
t
er
en
ar
o
es
O
"
e re
,
e
n
on
o
n e
.
s
o
r
e
n
n
es e
.
ct
an
st
t
e re o re
an
c
u
e
e re o re a
n
n
ro
s
o
s
a
a
n
e
.
ro
a
e e
O
s
s, u
a
n
e an
s
a.
e
o
ou
ac o r o
or o
o
as a x o
ac o r 0
er a
n
e
ra
ro
a
e
e,
es
r
s
c
ro
r
e
os
n
a
e
e r,
t
s
r
e
.
ro
e
e
ac o r
e
a
t
o
er a
s
o
,
e
o
n
e
o
a a re
a
on e
e re
e o
es
r
or a
e rs
u se o
n a
e
o r s n ce ( t
s
en
ar s e s o
er a
n
,
s n ce a
r
a
e
e
re
o
n
s a
n
o
a re s a
c
s
ac o r
e
a
e o
a
t
a
o n s, so
r
a
e
e rs
s
.
t
er,
er
an
o
e
r
er n
er
n
e
e n u
er
n u
n
e
ac o r ex c e
on
ro
os
co
a
r
t
o
te
s
a
e
os
co
e
ca
s
a
e
ex a c t
n o
s
s
c
s
co
n
an
se
s
c
.
e rs
an
a.
s
e
ro
s
r
ac o r o
s
e rs e
s
a
ac t o r o
an
e
o
e
r
t
e,
t
or e
e
can
a,
s
t
s,
a
e
r
e re
to
u
eac
.
c
e
a
c,
an
e
e
r
e rs
e n
o
ea c
s n o
e
on
.
to
or n o
e
c
e
e
e
to
eac
o
e
”
er
e
o
an
co n
a cto r o
e rs
n
e rs e
a
a
n
0,
,
s
er
.
c
r
s
,
e
o
an
HI GHER
34 2
AL
m
d b a re p ri
in t egral p o we r O f a is p ri e t o
i
ii
T is o llo w at On c e ro
( )
i
v
( )
If
an
a
h f s
v( ) I f i
l w t t m
a
o
e
el
mtb
409
Th e
Fo r if
e
r
s
mb
d
an
o
os
er
e
e
,
e in
g
an
i
l
t
u
f
q
i
m
i
l
l
t
u
u
p
q
o n s,
ra c
a
e e
us
.
o
t
e r,
e
n
e
e
er
ra
o
o s it i
v
e
fb
er o
n
Cb
e
d cl
0 an
h h v yp
i iv i t g l p w
eac
.
.
w
t
o
y
a re a n
to
e
.
m v yp t
f m
i
m
h
i
t
t
b
f
t
s
p
5
s, n
er
es
GE B R A
n u
mb
fp
er o
y
s it i
o
p
an
d
es o
fa
m
is
ri
es
v
in t e ge rs
e
g
db
re s e c
p
h
A ls o if 3
.
w tt
t iv ly
is in it s lo
an
in t
a re
es
e
e ir
2
m h
er
an
t
s,
d
en
.
m um
h
m m
h f
b e t h e gre at e s t p ri e n
b e r ; t en t h e
i
n
i
ch eac
3
5
ac t o r is a
ri
ro d c t 2
7
e n u
p
p
a n d t e re o re
b e r, is di i s ib le b y e a c o f t h e a c t o rs 2 , 3 ,
b e r o r e d b y a ddin g n it
t o t e ir p ro d c t is n o t
th e n
en c e it is e it e r a p ri
di is ib le b y a n y o f t es e a c t o rs ;
e
n
b e r gre a t e r t a n
b e r it s el o r is di i s ib le b y s o e p ri e n
i
s n o t t h e grea tes t
i
i
n e it e r c a s e
r
n
r
n d t e re
e
b
e
a
:
,
p
p
p
b e r o f p ri e s is n o t li it e d
o re t h e n
u
f
n o
t , le t p
on
ra c
wh
f
v
h
um f m
v
h
um
f
v
h
n u
No
o n ly
m
.
ra tio n a l
y
h
u
h
m um
m um
m
m
um
4 10
b ers
f
hf
m
a l e ra i ca l
g b
wh
en
m
a
t
h
at
is ,
a
o r
t
u la
fo rm
f mu
y
u
h
n
s
t p ri
m
e
p
n
dx
en
x =
re
p
re se n
a
e
mt h v lu
e
t
Of
:
h
e
t
p
e x p re ss io n
b ec o
m
es
m p ) (m p ) d ( m
mu ltipl
bm m
lm
m
l
i
l
f
u
t
p
p
p
b(
3
+ n
e
+ n
e
2
e e x p re s s io n
2
”
e
+ np
e o
a
e o
v
is di is ib le b y p ,
)
3
fp ,
,
an
d is t
h f
t
e re o re n o
m
ri
a
p
on
ly
e
er
411
wa y
.
b,
wh
A
.
n u
mb
be
er ca n
.
L et
a,
rep res en
can
2
0x
h wh
a
hu t h
u mb
h
h
.
um
x
m
.
la a
bx
I f p o s s ib le , le t t h e o r
b
e rs o n l , a n d s pp o s e t a t
ri
e n
p
t h e e x p re s s io n is p , s o t a t
m
h
u
e re a ,
a re
8
8
,
,
,
)
y
u mb
i n to p ri
u pp
i
m
u
m
b
u
S
pp
p
m
i
m
u
b
th
p
N de n o t e
0, cl,
res o lved
r
th e
n
er
e n
a re o
a
e rs
er
b eel
.
s
e
N
o se
o se a ls o
.
r
mf
e n
e rs
.
a eto rs
h
Th
t
at
en
,
in
N
=
h
on e
w e re
a8 3
, y
HIGHE R
344
n u
o
m
.
m
n
ca n
G E BR A
.
a s
i
n
h
w
w
i
c
h
a
f
y
fa cto rs which a re p ri
h
u
r
d
t
e
n
b
e
fi
b e reso lved in to two
To
4 14
b er
AL
o
co
m
p
e
o
o site
mt
ea ch
th er
.
Of h t w f t s
f l t th u mb N a
w u ld b
t h wis t h
mp w f
f
mu t t i
f t d m p w f i th t h f t
i
d th u
i
m
t
h
t
h
i
m
t b
S
il ly 6
h t w f t s w u ld
p
ly ;
f th f
d
H
th
mu t u i
h
l
t
t
m
b
u
u
u
u
f
w
y
h
h
m
i
i
w
i
h
i
b
t
d
q
q
l
i
w
b
d
t
t
t
u
v
f
s
t
h
i
h
t
t
d
t
;
p
wh
u mb
fw y i
i
u mb f difi p im f t i N
h
A
be
s
on e
s
n
on e
a
t
e
r
re
ro
s
er
e
s
a
fi
To
d the
um
u
1
(
+ a + a
v
2
di is o r,
ro d c t ; t
p
a
th e
u
su
h
a c o rs
e
r
mf th
o
m qu
re
h f
th e
e re o re
d
su
fa
o
n u
s
n
e
c
e
s,
e
a
e re
mb
n
s
e ac
h
er
.
f
b
as
m f th
o
a
1
er
th e
b
n
1
e o re
v
di is o rs is
e
h
T
.
en
e
u
q
a
h
l t o t is
umb
er o
m f th
o
b er 2 1 6 00
dv
dv
f i is o rs
i is o rs
e
1
c
um
.
3
th e su
en c e
.
1
C o n si
n
7
+ o + e +
i re d
.
on
ac o r
de n o t e d b y
e
s
.
divis o rs
e
Sin c e
th e
n
er o
ar
.
a
o
o
a c o r, a n
so
+
dt
at i s ,
m
o
e
er
er o
n
ac o r
z
an
E x a p le 1
t
su
b er b
L et t h e n
ro d c t
o f th e
p
m
ter
n
o
an
n
e
er
eac
on
o
e so
e o
o
e
or
eren
er o
.
re s o
o
n
e
e
t
s
'
e n
a
r
o
'
e re
a c t o rs
a
e
c an
er o
e
e
o
e
er o
o
e
n o
on e
n
n
4 15
is
so
c
er
o r o
,
o
c
n
a
ac o r
o cc
re
a n
”
ac o r an
o
s
t
co n
er
n
e
e
o re ,
q
3
2
(5
1) ( 3
26
1
2
—
—
63
1
x
2
3
1 ) (2
34 — 1
'
1)
3
5
3 —1
°
3
2
5,
72
—1
5— 1
40 >
<3 1
7 8 120
.
A ls o 2 1 6 00
or
4
w
E
be
vd
res o l e
ay s .
xa
We
m
pl
e
2
I f n is
.
hv
h m dv
Sin ce
on e o
c an
ft
a e
77.
e
o
dd
n
(
s
n
hw h
h
t
e
2
at n
l) = n
(n
(n
2
1)
1 ) is
e
.
o
e ac
h h
o
dv
t
i is ib le b y 24
er
.
(n +
u v v um h
h
u v um h
hm
v
du
is o dd, n
1 an d n + 1 are two c o n s e c ti
is i is ib le b y 2 an d th e o t er b y 4
e e
en
n
b ers ;
en ce
.
h
v
A gain n
1 , n , n + 1 are t
i i s ib le b y 3
IS
T
s t h e gi
3 , an d 4 , t at is , b y 2 4
dv
mt
in to two fac to rs p ri
hu
.
re e c o n s ec
en
ti
e x p res s io n
b ers ;
en c e o n e
is di is ib le b y th e p ro
e n
o
ft
ct o
e
f 2,
T HE O RY
m
E xa p le 3
d
Fin
.
h h
ig
th e
es t
U MBE R S
OF N
w
po
m
wh h i
dv
ic
f3
er o
345
.
tain
s co n
e
d
in
(
1 00
.
um
i is ib le b y 3 as th e n
an y are
b er o f
Of th e fi rs t 1 00 in tegers , as
ti e s t at 3 is c o n tain e in 1 00, t at i s , 3 3 an d th e in te gers are 3 , 6 ,
Of t es e , s o e c o n tain th e fac to r 3 again , n a ely 9 , 1 8 ,
an d t eir
by 9
n
b er i s th e q o tie n t o f 1 00 i i e
So e again o f t e s e las t
b er o f
in tege rs c o n tain th e fac to r 3 a t ir ti e n a ely 2 7 54 8 1 th e n
t e
b ein g th e q o tien t o f 1 00 b y 2 7
b e r o n ly , 8 1 , c o n tain s th e
On e n
m h
h
um
hm
d
m
u
u
u m
h h
h
dvd d
h d m
m
m
m
um
,
.
fac to r 3 fo r ti e s
H en c e th e i g es t p o er req ire = 3 3 1 1 3
T i s exa p le is a p art ic lar cas e o f th e t eo re
artic le
w
.
m
h
u d
u
h
um
.
1 = 48
mi
h
v
n
,
,
.
e s tigate
d
in th e
h
n ex t
.
fi
416
To
ta in e d i n
.
co n
n
f
h igh es t p o wer
th e
nd
o
m mb
r
i
p
a
t a in e d in
co n
re s
h
b e de n o t ed b y I
th e
co n
an
n
u mb e
ta in
d
so
a
o n
2
.
t
,
h
rs
at
T
wh i h
I
e re a re
a,
c
h
t h e h ig
t a in
mil
c
o
p
w
er o
fa
co n
m
on
a re
t a in
co n
a
p
3
v y
e c ti e l
t
h
en
g
le a s t
a at
h
re
t
e
y
wh i h
est
en a
a rl
dI
o n ce, an
en c e
co n
Si
2a, 3a , 4 a ,
le a s t
H
wh ich is
er a
.
L e t t h e gre a te s t in t e ge r
n
e n u
u
mb
o n c e, n a
e rs
mly
e
wh i h
I
c
le a s t
at
t a in e d in
I
n
o n ce
;
is
I
mi d
mu ltipl
417
I n t h e re
v e n ie n t t o e x re s s a
p
.
er o
a n
e
h
h
h
f t is c ap t e r we s a ll fi n d it
o f n b y th e s
b o l M (n )
4 18
To p ro ve th a t th e p ro du c t
divis ib le b y I
L
L
ym
f
.
r c o n s ec u
o
.
co n
ti ve i n tegers
is
.
u
Le t P" s t a n d fo r t h e p ro d
le a s t o f
ic h is n
t en
wh
h
(
PH
—
I
(
n
+
1)
(
ct
n +
n +
2)
P a
f
o
c o n s ec
2)
(
(
3)
n +
’
”
fi
r
n
u iv
t
e
in t ege rs , t h e
+r
(
n
)
+ r ;
+ rP
P
n
n
r
m
ti
es
u
t h e p ro d
ct o
fr
1
c o n se c
u iv
t
e
in t e ge rs
.
H IGHER
346
H
hv
1,
r
u
if t h e p ro d
we a e
en c e
ct
f
o
AL
GE B R A
1
c o n se c
P
= rM
r —
t
(|
u
h f
u iv
.
v
in t ege rs is di is ib le b y
e
1)
r
mu lt ipl
h
f
I
|
mu ltipl f [ W h v hu p v d t h t if
l
P P
1
u tiv i t g i di i ib l b y | 1 t h
h p du t f
i
i
i
i
i
i
l
d
u
u
v
v
b
b
f
t
h
d
t
b
t
;
g
y E
p
w
i
t
i
t
t
i
di
i
i
l
u
v
y
u
v
v
b
d
t
f
b
;
g
p
y l
g
th
f h p du t f v y t h
u tiv i t g i di i ib l
ll
by
y
d
g
3
L
my l b p v d hu
Th i p p iti
f A t 416 w
h w t h t v y p im f
By m
l
i i
f
t i
i
d
i
t i d i
I
L
:
i
t th
tud t
Th i w l v
N ow
a so
t
3 ,
ro
c
o
r
ro
c
o
e
t
e re o re
an
ro
s
a n e
s
o
e
ro
c
so
on
e n e ra
r
th e
ffi i
c en
t
h th
o
f th e
t en
ro
c an
at
e
e re o re
e
a
r
s
e
t
e
s
o
m mb
f m
ro
t
e
n
s
e
e rs
s
e
,
e
e rs
u
e
v s
e
s
o
e
a
t
ea s t a s
e x e rc s e
e e x c e p t io n
r
s
s
e
n
f
v s
s
s
e
e
e
,
.
W it
e rs
e rs
t
e
ree c o n s e c
a so
.
e a s an
ea
e
a
o
.
r as o
n
n
te
er
a
on
n
e
e
c o n sec
e
n
_
a
.
e
e
o
er
If p is a p ri
exp a n s io n of ( a
419
e
e
t
c o n s ec
o s
r
c o n s ec
e an s o
s co n
es o
r
o
c
P2 is
e re o re
a re
4,
ro
e
d t
r, a n
e s
th e
er
s co n
en
r
a n e
ac to r
e
n
r
n
.
m
in
efi
f
ex c ep t th e firs t a n d la s t, is di vis ib le b
y p
er,
e n u
co
c ien
t
ter
every
o
.
o
f th e
fi
rs t a n
d la s t ,
e
v y t mh
er
er
as a c o
or
— l
)
r(r
(
— 2
r
)
( tr
If
wh
thi
my h v
v u
'
i
n t e ra l
a l e n o t e x c e e din
w
1
N
o
y
g
gp
e n o
ac to r o f
s e x re s s io n is a n in t e ge r ; a ls o s in c e p is p ri
r
p
(
is a di is o r o f it , a n d s in c e p is gre a t e r t a n r it c a n n o t di ide
r ; t a t is ,
1
r
1
s
acto r o f
+
t
an
b
e
p
p
p
(
(
)
y
(
)
H
e n c e e e ry c o e f c ie n t e x c e t t h e
di is ib le b y I
rs t a n d
p
I
e re r
v
f
a
a
v
e an
v
v
t h e la s t is di is ib le b y p
.
W rit e
B
If p is
fo r b
a
p
+ c +
fi
fi
e n u
t
h
mb
en
er,
to p ro ve th a t
b y t h e p re c e din g
s
h
mu
v
.
m
ri
.
f
.
.
4 20
m
.
h
h
I
‘
B y p ro c ee din g in t is way we
my
a
e s ta b
lis
u pp
h th
i l
a rt c e
o se
e re
;
u
i
q
re
d
re s
ul
t
.
HI G HER
34 8
AL
E XA M PL E S
1
d
Fi
n
.
v
Fi
n
.
e rs o
wh i h will mk h
d h l mul ipli
t
t
eas t
e
t
e
a
c
e
X XX
.
f th e
n
p
ro
du
a
.
.
e rs
7 4 08 8
e rfe c t s
p
q
c ts
f th e
e rs o
.
u mb
1 83 7 5,
3 6 7 5, 4 3 7 4 ,
re s p e c ti ely ,
2
multipli
th e lea s t
GE BR A
n
u mb
u
are s
.
e rs
7 6 2 3 , 1 09 3 50, 539 539
v
res p e c ti e ly ,
wh i h will mk
a
c
v
I f x an d y are p o s it i
2
z —
i
s ib le b y 4
i
s
i
x
y
3
is e
v
4
dv
Sh w h
.
en
t
e
.
I f 4x
6
Fi
.
n
d
i
s
y
th e
ub
es
y is
x
.
e
v
en
s
,
hw h
e
t
at
multipl
f 3,
e o
a
u mb
dv
an
ay s
c an
t
2
th e
n
4 x + 7 xy
at
f 8 06 4
o
e r an
an y n
e
f i is o rs
er o
my w
s
u mb
e en
d it s
2y
2
sq
u
are
d i i ib l
is
v s
e
.
umb
er
7 056 b e
vd
l
re s o
e
in to
i d ivi ib l b y
v h
h t ( ) ( ) i mul ipl f
P v
9
b
S h w th t v y um
d it
ub wh d ivid d b y l v
10
m mi d
If i
v hwht (
d
v
11
i
i
i
ib
l
b
)
y
S h w th t
i
v
i
l
i
d
i
b
b
12
)
(
)(
y
t
If
h
hw h
i g
i di i ib l b y
13
8
.
Pro
e
t
at
2
.
ro
e
t
a
n
e re
sa
.
Pro
t
.
o
.
e r an
n
t
s a
s c
n
t
e
2
a
1
t
n
n
2
+ 20
um
18
19
.
3n + 2
2,
an
s
s
.
3
+ 7 is
r
at
6
e
ea
e
t
e
at n
e
e o
a
er
n
f8
re a e r
24
e
5—
48
e
.
.
3
5n + 4 u
vs
s
e
.
t
h
an
3,
s
hw ht
t
e
a
n
2
1 is
5— n
n
is
d i i ib l
v s
e
b y 3 0 fo r
all
v lu
a
es o
f
n
,
an
d by
.
h
d
h
dv
e
t
a
n o s
are n
a
e
c
er
e n
sq
u
ares
o
f
an
t
w
o
y
m
p ri
.
h w th t qu umb i
Sh w h t v y u b
u mb i
e
.
en
s
s
multipl
b
t
g
p im u m
at
2n
dd
h
S
e
s
iffe ren c e o f t h e
S e w t at t h e
17
b e rs greater t an 6 is i is ib le b y 2 4
.
6
e o
.
h w th
e
s
n
a
.
S
16
2 4 0 if n is
,
rea e r
v h
1 5 I f n is
u l tiple o f 2 4
m
en
s
e
er
+5
n
15
e
s
s
+1
n
e
a
n
.
1
.
s e
e
.
4n
er
a n
n
.
14
a
e
.
n
n
w
hw h
th e diffe ren c e b et
at
.
a
d if
an
c
.
In h o w
7
t wo fac to rs ?
th e
p e rfec t
c ts
in t egers ,
e
.
5
du
th e p ro
e
er
s o
er
m3
f m9
f th e fo r
s o
f th e
or
77.
1
.
n or
9n
i
L
.
e
,
T HE O RY
S
20
.
.
is 0, l
hw h
6
or
t
e
if
at
a c
u b umb
e
um
.
22
.
S
.
hwh
t
e
.
re
mi
a n
d
er
d
an
are
ub h w h
e, s
c
t
e
it is
at
o
f th e
h w th
e
at a
e
be
er,
hw h
m u mb
d iff t mi
r
i
p
a
umb
er c an
n
x
e
n
re
e ren
a an
d
a
a n
s
d
m3
f th e fo r
o
1
.
t at 1
e
e rs
2
2
,
wh
2
,
en
.
w y v wh t v
a are al
”
6
a
a
s e
en
a e
,
e r a an
d
x
.
Pro
25
f m
.
8r + l
or
26
27
1 7 7t h 1
at e
v h
at
t
v
t
e
v th
e
th e 8
at
v
S h w th
o
.
w
er o
f
e
v
ery
dd
o
n
u mb
is
er
o
f th e
w
er O f an
y
n
y
n
u mb
is
o
f th e fo r
u mb
is
o
f th e fo r
er
m1 3
77.
“
o
p
wer
f
o
an
er
ml 7
n
.
e
n
umb
er
at n
36
er
e n
t
h
an
r
ea
t
e
r
g
t
h
greater
m u mb
If n is an y p ri
l is di is ib le b y 1 6 8
30
an d 3 7
o
p
po
m
.
en
.
.
29
v
th e
I f n is a p ri
28
i is ib le b y 2 4 0
dv
e
e ry
.
Pro
.
v h
e
Pro
.
1 3n + 1
or
s
u
v
S
24
ay b e
or
h qu
t rian g lar
at n o
.
m
b y 7 , th e
e
.
I f 2 n + 1 is
23
di i ed b y 2 n + I lea
vd
d ivid d
is
er
n
3 49
.
.
If a n
b er is b o t
21
or
7 77 o r 7 n + l
f m
UMBE R S
OF N
5,
s
3,
an
hw h
t
e
e xc e
at
t
p 7,
4
n
s
— 1
h w th
e
is
at
.
1 is
d i i ib l
v s
e
b y 3 3 7 4 4 if
mt
is p ri
n
e
2, 3 , 1 9
o
.
h
d ivi
W
l
QP —
x
l is
2p + l
an
en
s1
le b y
d 2p + 1
m
h
bot
i
n
r
e
p
if x is pri
a re
umb
mt
s
e rs ,
e
o
hw h
t
e
2, p + 1 ,
at
d
an
.
I f p is
32
"
di is ib le b y p
v
.
33
p
ro
is
If
mi
v th
multipl
.
s
at
e
a
e o
4 23
m
e, a n d x
r
i
p
a
a
mt
p ri
e
o
e r, an
d
a,
m umb
p ri
e
n
p,
f
h w th
b t wo
4 2
m
m
3
m
6 +
a
i
b +a
M a
e
nu
at
mb
e rs
r
— 1
P
les s t
is
h m
an
,
m
2
+b
-
-
-
s
xp
“
m
.
um
h
h
um
m
I f a is a n y n
b e r, t e n a n y o t e r n
b er N
a
y
b e e x p re s s e d in t h e o r
N
a
re
r
e
i
h
i
n t e ra l
s
t
e
+
,
g
g
o
i
t
e
n
t
e n N I s di ide d b
a , a n d r is a re
a in de r le s s t a n a
q
y
Th e n
b er a, to
ic t h e o t lie r 1 re e rre d, is s o e ti e s c a lle d
the
o du lu s ,
an d t o
an
i
en
o
l
d
s
a
t
r
a
r
if
f
e
e
e
a
d
r
e
n
t
e
y g
u
.
wh
um
m
f m
v
wh h
:
wh
s f
v muu
q
m
h
m m
h
.
HIGHER G
m
i
m
d
t
i
f
f
u
h
f
b
N
d
t
f
m
p
g
f
v lu f Thu t mdu lu s 3 w h v u mb f th f m
m imply 3 g 3g l i 3g 2 i
1 3g 2 ;
3g 3 9
i
m
l
u
I
l
k
m
t
d
u
u
5
u
1
1
m
l
3
b
t
g
y
)
q
(
w ill b
f th fi
f m 5g 5g 1 5g 2
w i t g
If 6
wh i h wh di id d b y
4 24
m mi d t h y
id
t w it h
l v th
b
l
h
m
d
I
t
i
i
t
t
u
u
h
b
m
u
ltipl
f
d
p
f ll wi g h t ti f G u w h ll mtim p
hi
f ll w
3 50
or
s
a
e
o
r
o
,
er
n
a
+
+
,
as
e
sa
ec
o
t
n o
6
E it
on
a
h
ft
er o
h
E e
ese
,
o
o
ss
a
m
d
(
f mu l
o
.
)
a
b
s
.
to
s
so
a
co n
e
r
e
e
o
e
es
a
.
)
a
gru en
e o
m
d
(
0
c E
s
n
v
s a
c
or
+
an
en
sa
e
.
c
e
or
,
is
a
e
o r
4
e ren
a
s n ce
,
o
s case
n
.
o
,
a re
e
o
e rs o
i
e rs ,
e
n
e r,
s a
s
o
t
4
n
n
,
an n e r
s
on
e
a
,
e
or
a n
o
e
s
n
a re
.
re
e
e
t
n
o
e
,
e
o
o
o re
ve
e
,
o r,
e
.
re s
o
.
e on e o
ea
o
s
.
.
c o rre s
or
e ac
,
o
a
e
E BR A
AL
ex
a, an
t
re s s
s
.
c a lle d a c o n gru en c e .
md
If b , c a re co n gru en t with resp ec t to
4 25
b
a n d p c a re c o n gru en t, p b ein g a n y in teger
p
o
.
u
lu
th en
a,
s
.
Fo r, b y
t
h f
b
p
e re o re
4 26
s
o s it io n
=
c
p
If
.
u pp
a
p
n a
is p ri
divi ded b y b , the
a re
m
'
Fo r if p o s sib le ,
a
wh
v
s
w
;
e
b,
o
mi
re
u ppo
th e
e
m n I—
m
m
(
)
a
a ll
a re
t h at t wo
se
v
1)
b
(
ders
a n
m
in t e ge r
e
.
d the qu an tities
an
2a , 3 a ,
di ide d b y b le a
en
v
mt
a,
h
w e re n is s o
b
n a,
c
h ic h p ro es t h e p ro p o s itio n
,
sa
a
diferen t
.
u
f t h e q an t it ie
re
a in de r r, s o t
o
m m
e
sm
ht
an
a
d
a
r,
-
h
6
(q g ) ;
th
f b divid (m m) h it mu t di id m m i
i i p im t
ib
l
i
m
h
d
m
; b t t h is i im
p
l
b
th
Thu t h
mi d s
ll diff
f h
t
d i
i
t
i
i
l
i
t
di
i
l
b
h
u
y
v
b
u
b
m
t
b
h
m
d
i
t
q
y
i
ms f th
1 2 3
s ily i t h i
b
1 b t
t
t
’
en
e re o re
t
s
an
ter
o a
u
a
—
:
en c e
s
v
s
o ss
e s n ce
e
an
'
'
s n ce
,
a re
eac
.
s
t
’
es
e
r
ess
an
a
-
'
e
s
es
re
a re a
ex a c
s
e s e r es
o
er
a n
,
,
e re n
e
e re
,
,
u
,
s n ce
an
,
a n
none o
s
e rs
n ec e s a r
n o
e
t
e
t
e
s
n
o rde r
.
C OR
o f th e A
If
.
.
P
a
mt
is p ri
e
o
b,
an
d
c
is
an
y
n
um
be
.
c,
c
+ a,
c +
2a ,
c
+
b
(
—
l)
a,
r,
m
th e 6 ter
s
HIGHER
3 52
s h ew
To
4 30
ea ch o th er,
.
if
th e
a
ut
h li
ab
c
+
;
t
n e co n
b,
ers a ,
d,
c,
h
fi
th e
en
a re
t a in in g
u mb
u mb t hu
rs t
a n
1,
a +
2a
1,
+
p ri
mt
e
o
b
n
e rs
can
s
k)
2,
h,
a +
2,
a + a,
h,
2a
2a
b
(
b
(
b
(
a
e rs
2,
1,
2a
mb
.
b cd
C o n s ide r t h e p ro d
rit t e n in b lin e s , e a c
a +
n u
G E BR A
<
4
w
th a t
AL
+ a,
b
(
u m wh h
h
m h s um
m
m
mm v
v
um
um
m
fi
um
h f h
v
um
h
m
m
h h
y m
u
v
u m wh h
w h
h
um
m wh h h
v
v m
um
s
v
n
ic
b egin s wit Io if
Le t u s c o n s ide r t h e e rt ic al c o l
n will b e
ri
e
t
o
a
k is p ri e t o a all t h e t e r s o f t i c o l
p
;
o n di is o r, n o n
b e r in t h e c o l
b u t if k a n d a h a e a c o
n
N o w t h e rs t ro w c o n ta in s qS ( a ) n
b e rs
will b e p ri e t o a
e rt ic a l c o l
n s in
t
o a ; t e re o re t e re a re qb ( a )
ac
ri
e
e
p
is p ri e t o a ; le t u s s pp o s e t h a t t h e
ter
e ve r
o f w ic
ic b e gin s it h is o n e o f t e se Th is c o l
n
n
e rt ic a l c o l
ic w en di ide d b y b le a e re a in de rs
is an A P , t h e t e r s o f
b — l [A rt 4 2 6 C o r ] h e n c e t h e c o l
n c o n t ain
0, 1 , 2 , 3,
e to 6
i
n t e e rs pri
)
b
(1
g
( )
.
.
.
.
m
m y
m m
.
.
.
h
u ms i
v
i
a
e
r
t
a
l
l
c
c
o
Si ilarl , e a c o f t h e <
)
1( )
is p ri e t o a c o n t a in 4) ( b ) in t e ge rs p ri
ter
t ab le t e re a re
4) ( b ) in t ege r w ic
t at is
a l o t o b , a n d t e re o re t o a b
h
s
h f
a
>
b
<
l( )
T
h f
e re o re
f(
( )
abc
m
s h h
h
=
n
m
95( a )
a
5
q ( )
d
h h v y
w ic e e r
e to b
h e n ce in t h e
a re
ri
e
t
o
a
an
d
p
n
<
1:
b
ed
(
M
5
9
ed
M
d)
431
i
g ven
wh
.
n u
To fi n d th e
b er, a n d p ri
m
L et N
e re
o
s
i
t
i
p
b e rs
v
a,
mt
c,
in t e ge rs
”
1 , 2, 3,
a
e
.
a,
e
mb
o
er
f
o
o
s
i
t
i
v
e
p
in tegers less th a n
.
u mb
diff
n
2 a , 3a ,
a
it
h
d s u pp o s e t at N = a c
b e rs , an d p , g,
a re
e re n t p ri
e n
”
o f t h e n a t ra l
C o n s ide r t h e a c t o r a
”
1 , a , t h e o n l o n e n o t p ri e t o a a re
de n o t e t h e
b,
nu
e r,
an
f
y
m um
— 1
m
s
)
a,
(
u
p
—1
)
a,
’
u
'
r
n u
m
T HE O RY
an
d th e
u mb
n
Now
er o
a ll
f
the
h i
4w
ft
ese
s a
a c to rs a
OF N
h
z
—
” 1
,
,
)
<
1(
p
1
a re
,
1( )
i
u
( )
( )
3 53
.
en c e
'
"
b c
”
U M BE R S
"
b
)
p
mt
ri
e
o
eac
h
an
r
e
is
x
S
o
it is
t at th e
§ N ¢ (N )
1
’
'
"
b c
su
e
mf
h
y in teger les s t an N
t an N an d p ri e to it
m
h
an
mt
d pri
e
o
wh h
h
ic
it , t
en
are
N
-
h
le s s t
is
x
an
als o
N
an
.
te th e in tegers b y 1 , p , q ,
en o
th e in tegers
all
o
.
an
in teger le s s
D
hw h
.
e
xa
If
;
1
m
pl
d p im t
E
er
”
is ,
at
t
( )
1
c
( )
e
t
o
'
1
a
h h
an
r,
h
dt
eir s u
mb y S ;
h
t
en
+ (N
th e
serie s co n s is tin
Writin g th e
ro
s whi h
c
N
h
v
d
re ers e o r er,
a re
N
s
la t a rt ic le it
le s t h an N an d
e
s
m;
)
ter
s
f ll ws t h
o
n o
o
at
mt
t p ri
e
o
th e
n
it is
u mb
o
f
er o
f
er
1
l
is ,
at
H
s.
s (N
to a
F mt h
.
in t e ge r
t
ter
ditio n ,
a
4 32
in th e
s eries
m
)
S = (N
d
by
g o f 95 (N
e re
ma g v
N
t h e ter
i
c
H
co n
.
H
.
t a in
A
.
a
as
a
th e
n
u mb
2a , 3 a ,
a,
wh i h
es
f
a c to r
h
e
t
;
er o
f th e
In
t e ge rs
N
.
a
a
m
te r
i
g
v
es
th e
n
u mb
HI H E R
3 54
AL
G
s
t h e in t e ge r
a
N
b , 2a b , 3a b,
GE BR A
.
h h
w ic
a b,
co n
t a in
a
b
as a
f
a c t o r,
b
in t e ge r is re c k o n e d o n c e , an d o n c e
rt e r, e ve r
an d s o
on
lt iple o f a b will a pp e a r o n c e a o n g t h e
s, eac h
t
onl ;
lt iple o f b , an d o n c e n e gat i e l
lt iple s o f a , o n c e a o n g t h e
s re c k o n e d o n c e o n l
lt iple o f a b , a n d is t
a
o n g th e
N
N
N
lt iple o f a b c will app e a r a o n g t h e
A ga in , e a c
,
a
0
b
ltip les o f a , b , 0 re sp e c t i e l ; a o n g t h e
ic h a re
t er s
y
hu
mu
m
a
b
an
’
ao
h
mu
s
mu
m wh i h
ter
be
es o
e
on
c as e s
4 33
se
sc
e
a
c o cc
a
l
so n
i
W
[
.
is divisib le b y p
’
m
m
h
a
b,
ac
,
is ,
at
o n ce o n
be
re s e c t i
p
s in c e
ly
3
3
mil
Si
.
v ly ;
e
1,
l
ly ,
ar
o
1
I
t
h
er
s Th
eo re
m)
If p b e
.
a
p
ri
m mb
er,
e n u
1
p
hv
a
.
e
< >
— 1
p
=
F mt
d by
er
(p
a
’
h
T
s
(p
eo re
is
m;
) (p
t er
m h
ea c
o
f th e
or
1
o
1
M
—
(P
O)
M (p )
U
h f
Th i t h
f t
ere o re
s in c e
1,
I
(
p
—
mi
p
—
l is
e
v
p
(p
re s s io n s
t
hu
s
1)
s
—2
1
P"
D
ex
f m
f th e
-
T
t
d
.
.
—2
an
abc
f
v y
y
.
.
I
f
rs o n c e , a n
By E x 2, A rt 3 1 4 we
— 1
p
hu
v y
c
a
e o
s
es o
m g th b mu ltipl
h mu ltipl f b u
my b di u s d
a
m
mu ltipl
a re
c
s
N
d
eac
y
mu
m
mu
h mu
m wh
’
Fu
.
a
m
l ter
—
s
)
en
.
l
y u w h p i p im F u pp p
h
q t h g i l t h p d mu di id ip l h
1 i
I
t
mu ltipl f g d t h f
t
ml ipl fp
Ip
W il
Th
m my l b p v d w it h u u i g th
su lt qu t d f mA t 3 1 4 i th f ll w i g ti l
s
as a
e o re
s
or
ac
en
s n o
’
so n s
re
o
e
on
l
tr
s
es s
a
e o
e o re
a
ro
r
.
e
en
an
,
,
as
st
an
ro
e
n
v
e
o
o
n
en ce
e
a
o
e
o se
or s
.
e re o re n o
an
a so
e
r
s
ar
t
u
t
c e
e o
s n
.
.
e
HI G HER
3 56
y h
AL
G E BR A
.
m
M an t e o re s relat in g t o t h e p ro p e rt ie s
4 35
b e p ro e d b y in d c t io n
.
can
E
xa
L et
v
m
pl
1
e
0
x7
x
u
.
d
be
f(
en o
x
te
d
m umb
by
f(
x
0
er, x 1
e n
h
);
t
x
dv
is
—
i is ib le b y p
(
—x
xp
—l
r (p
)
h
If t
e o
an
h
d t is is
h
P
=
2
2
f( )
wh
dv
h
mu
v h
d d
m
h u h
w h
2’H ‘2
h
Pro
.
e
at
t
25b e
2 4n
Let 5
t
‘
2 W+2
en o
by
te
27 “M
5
f (n
en
— 52
f(
n
+ 1)
’
t
2 5 is
m [A
p ri
s
e
rt
.
.
but
t
h
dv
eo re
h
erefo re
d
f ( 3 ) is
th e p ro p o s itio n
s
mf
or
,
ivis ib le
is t r e
u
mt
if x is pri
i is ib le b y 57 6
e
o
p,
.
f (n ) ;
(n
24
1)
2 4n
5
2 5f (n ) = 2 5 ( 24 n + 2 5)
dv
u wh
s
.
2 n +2
57 6
h
a
2 4n
5
.
,
hu
.
mt
T is f rn is e s an o t er p ro o f o f Fer
l
p
ltip le o f p
1 is a
it fo llo s t at x
m
—2
e [A rt
en p is p ri
fp
an d s o o n ; t
s
i
b
l
b
4
i
e
i
i
s
,
y
p
f( )
.
e rs
)
is f ( x +
s o als o
eo
E x a p le 2
,
e o
v
x
multipl
a
b y p , t erefo re
n i ers ally
u v
+ 1)
is di is ib le b y p ,
f( )
e refo re
u mb
m + 1”?
multipl fp ifp i
multipl fp
a
f(
n
en
+ 1)
x
f
.
a p ri
I f p is
o
25
49
24 n
49
(n
T erefo re if f ( n ) is i is ib le b y 57 6 , s o als o is f (n +
b u t b y trial we
en n : 1 , t e re fo re it is t r e
is t r e
en n = 2 , t ere
s ee th at th e t e o re
s it is tr e
an d s o o n ; t
n i ers ally
fo re it is tr e
en
Th e
h m
u wh
v u m
ab o
e re s
ay als o
lt
2 4n
b e pro
25
25
n
hu
vd
h
as
e
+1
2 4n
—2
5
M ( 57 6 )
E XA M PLE S
hw
Sh w
Sh w
Sh w
S
h
h
h
h
e
t
e
t
e
e
t
t
at
10 + 3
at
2
7 +3
4
"
6
8
"
at
at
n
7
.
5
1
25
— 2 4 n —2 5
.
.
XXX
.
d ivi ib l b y
i
mu l ip l f
wh divid d b y l v
5
en
is
o
.
25
n
is
n
h
s
— 2 4n
— 25 2
+ 5
n
fo llo
u wh
u u v
w
s
t
s a
e
9
e
24
e o
20
m24 (2
f t h e fo r
.
r
ea
.
e s re
mi
a n
d
er
9
.
Y
TH E OR
m
I f p is p ri
5
.
6
7
2"
8
9
1
hwh
Sh w th
P v h
ro
.
in
t
o
at
t
t
e
3
+5
+ 1 6 0n
th e
o f (1
p
is
2n + 1
2n + 5
v
o
f 2
I f p is
.
f (1
tip le o fp
2
o
e
n
a
e c o ef
fi c ie n
e
s
ts
wh
gre a te r t
er
u mb
p ro
e r,
n
are
m um
h
v h
t
e
h w th
b e r, s e
p ri e n
alte rn at ely great er
2
is
1
o
en
b y 51 2
.
f th e o dd p o we rs
n is a
ri
e n
p
Of x
m u mb
er
an
7,
at
6
s
hw h
t
e
at
n
G—
l is
n
4
2 —
n
n
3
11
+
+7
is
a
at
th e
S
,
to 9 1
d les s b y
an
17
at
m h
hm
hwh
e
t
.
n
18
I f p is
.
“
a
at
b
lz
.
m h
p ri
a
—l
,
Z—
4 ) is
(n
If
n
n
ity t
h
o
m
m ml
f th e t e r
an
so
e
s
u
m f th ( p 1 ) p w f
wh i th mm d iff
h multipl fp
b y 9 1 if
d b
b t h p im
h
v
is di is ib le
e, s
+ 1 are
n
e
ul
a
v s
t
e o
t
e
a
e
o
‘h
e
e re n
a an
e rs
o
e co
e o
,
on
o
er
.
are
o
r
e
20
.
2r 2r
a
30t +
or
es t
po
w
er o
1 is
1
d ivi ib l
h
s
greater t
2
2
n ( n + 1 6 ) b y 7 20
12
e rs
n
.
by p
e
5,
s
A ls o
s
an
.
hw
hw
e
e
.
fn
wh i h i
c
s co n
tain e
di
n
I
n
f
—1
— n r+ r — 1
to
1
n
m
d ivi ib l
Fi d h g
1 is
2
3 0t
e
.
1)
htp |
t h p im u m
b
t
bo
r
e
b y 1 2 0, an d
or
u mb
ud uh h
If p is a p ri
19
b e r c2 c an b e fo n
um
ew
d i i ib l
mu b f h f m
Sh w h t th h igh
st
at n
-
u
ts
.
16
18 eq
c o ef
fi c ie n
.
m
.
15
a
in
.
n
O dd
dv
loo
d
.
d ivi ib l
2 4 3 is
56 n
m f th
su
f 14
e o
I f p is a p ri e , s ew t at th e s u
an
t
i
c al p ro gre s s io n ,
n u
b
e
r
s
i
n
a
r
i
t
e
y p
e n ce is n o t
i is ib le b y p , is les s b y 1 t an a
14
n
e
"
.
o
t
ta in
co n
multipl
a
m u mb
ri
an
.
13
h
h
.
.
I f n is
12
ltiple o f 1 2 8
t
fp
e o
.
er
o
mu ltipl
a
b y 30
e
s
es t
4n + 2
at
a
.
mu
3 + 1 is
2 p
d ivi ib l
h igh p w
5, is di is ib le b y
an
If n is
11
di is ib le b y 504
v
3
at
e x pan s io n
h h
er
t
e
o
.
(
at
is
a
th e
at
t
e
4b + 1
at a
t
hw h
3 57
.
.
S
.
10
th e
t
e
o
.
r—
-
hw h
Sh w h
S
.
e, s
UM B E R S
o r N
n
s
t
e
e
e
s
by p
d
mt
p ri
2—
t at 0
a is
e r, an
n
c
a
dv
o
.
e n e ral s o
9 8x
u
l tio n
1
5
o
0
f th e
d
m
(
o
.
u
h h
p , a n d if a s q are
i is ib le b y p , s e w t at
e
u
c o n gr e n c e
H I GH E R
3 58
a
GE B R A
h w t h t t h m f th q u
u mb N d p im i i
S
21
n n
i
e
g
v
AL
.
e su
a
e
t
to
e
r
an
er
e s
o
ares o
.
f all th e
s
— a
an
d th e
su
m f th
o
e c
ub
b ein g th e differen t p ri
If p
22
di is ib le b y
an
.
v
d q
g
(
h w th
v
u
h
t
b
i
q
y
g
S
23
.
at
e
e s
en
p an s io n
o
f
are
an
th e
are s
2 ’
6x + x
n t ago n al b y th e c o ef
e
p
1
an
mf
u
sq
v
i
t
i
o
s
t
w
o
y
p
|p
are n
th e
an
h
an
)
u mb
ffi i
in t ege rs ,
e
h h
s
h w th
e
are als o
c en
ts
o
f th e p o
th e
sq
u
are
h
at
ts
o
f th e p o
c en
.
w ic
ers
co e
)
— b
1
1
)
(
(
s
i
p
g
(
at
.
d t
fii
fN
ac t o rs o
e
d by
of
les s t
e rs
— b
1
) (1
(
—a
4
b,
u mb
is
es
N
—
a,
n
w
ers o
ers
ers
w ic
w
u mb
n
u
t rian g lar
o f x in t h e
h h
f x in th e
are
e x pan s io n o
are
ex
als o
f
1
1 — 1 0x + x 2
h
h
'
m
h
les s
.
m
A
?
3
mf
b ein g th e differen t p ri
b,
25
m
.
e
umb
If <
N
i
n
s
h
t
e
)
)
(
1
to it, an d if x is p ri
mt
26
.
hw
e
d
I f dl , d2 , d3 ,
als o
x
t
h
en o
e
e
o
1
hw h
E
te th e
0
m
d
(
o
divi
.
s o rs o
I
at
x
5
th e
n
b
3
e rs
1
(
.
are
les s t
h
an
mb
N, t
h
en
.
fa
u mb
1 — b)
3
h h
N)
o f all
>
<
f in t e gers w ic
N , s e t at
( )
3
x
a
fN
a c t o rs o
er o
’ lN ’
d
x
S
1
(
ers
a
-
30
p ri
i
<
—
N
a,
w
o f th e fo u rth p o
S e w t at th e s u
t an N an d pri e t o it is
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9
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ffli g p k f
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11
A h
3 h
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l
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h i l y t i i g p i d b l k mp
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12
Sh w th t h
h
f h wi g i wi h
d
i
6
p tiv ly
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h
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t
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o
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2
S
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c
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8 to 3
e c
an c e o
e
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t
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er
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t
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a ro
ers o n s S
,
t
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n
ac es a
ar
t
en
e o
r c ar s are
t
e
S
ares
on e
a
n
tt
to
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et
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5
o
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2
e
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er
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s t,
c
a a n st
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an
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:
n
o
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a
1
e
c ar s ,
c ar s S
n
ro
4
n
t
8
e,
o
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a
tter
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are as
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at
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t
at
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; t
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n
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ta
ers o n s
as
.
ec
ac
are
12
l c e , an
res
r
s a a ns
e c
at o
a
.
t o ss e
ts
t e
e re are
o n e c an
t
t
n
.
en
o n e a a n st
.
t
ra
s
o e
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IS
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r z es an
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9
r z es an
6
e
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s : co
s
;
are
.
c
an c e s
18
.
o
n
S x
t
4 , 3,
o r
2
ce
R O B A B ILI T Y
37 7
i i g f v lu m
h w k
Th
f
13
h f v l m Th y p l d
h lf
dm
d h
l
m
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m
v
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P
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e
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o r s, o n e co n s st n
ree
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3
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es , o n e o
S
at ran
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;
et
er
3
is
1 40
14
w wi h
h h u mb
A an d B th ro
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h w
ft
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.
.
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ro
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ce
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9 , fin d B
s
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d mi
or
Clifto n are p lac e at ran
1 5 Th e lett ers fo r in g th e
o
e ls c o
at is th e c an c e t at th e two
e t o ge t e r ?
ro w :
.
h
h
h
vw
h
h
m
o
h
n
h d wh i t wh t i
th t h
ki g
h d
fi d pl y ?
h illi g d h lf w pl d
d mi
Th
17
f th
m i b i g b h h lf
hw h h h
li
1
w i 7 G li h i ul i h
f m h illi g
d
h lf w
I n a an
16
e l b y a s p ec i e
at
.
.
a
ne :
S
ns
s
c ro
a
n
4
t at t
-
e
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t
ze
3
an
s
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t
s
er
a
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a
s
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a
o
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re s
t
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-
ns
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t
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O
S
are
o
n
t
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a
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an
s
.
v
h
h
s
y h
u
m v t
w h h h
4 55
W e h a e h it e rt o c o n ide red o n l t o s e o c c
ic
in t h e lan g a ge o f Pro b a b ilit a re c alle d Si p le
W h e n t wo o r o re o f t es e o c c r in c o n n e c tio n it e a c
t h e o in t o c c rren c e is c a lle d a Co p o u n d e e n t
.
wh h
j
s
n
at ran
ace
c as e
4
e
u
m
u
mpl
u
y
m
v
rre n c e s
e
en
o
t
s
.
e r,
.
u pp w h v b g t i i g 5 wh it
d tw d w i g
d 8 b l k b ll
h
f th
b ll
iv ly
I f w wi h
md f mit u
t im
h
th
3 b l k b ll w
f d w i g fi t 3 wh it
d th
h u ld b
mp u d v
d li g wi h
igh t
uh
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th
d d wi g m
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th
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3 wh i
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d
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hus h h
f d wi g 3 b l k b ll
th
u lt f t h fi t B if t h b ll
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ff
d b y th
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fi
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y
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t h u l d t t h f ll wi g d fi i i
W
d p
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id t b
Ev t
f h
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f
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t ff
th
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th
D p d t
Fo r
ex a
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S
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W
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s eco n
ac
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s,
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t
t on
o
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a
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s a re s a
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en
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n
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en
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s,
a
e
ca s e
e
ra
.
e
ac e
re
re a t e r
s
a
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te
e
o
a
o
c c es s
a co
c
e
an
rs
n
n
s,
s
n
a re n o
s
a
ac
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s
e,
a
re
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rs t
e
.
on
e
ect
sa
en
en
e o cc
o
acco r
n
rre n c e O
e co n
t
n
en
as
t
t
.
e
37 8
H I GH E R
If th ere
4 56
b a b ili ties
.
a re
wh ic h
f
o
a re
AL
GE B R A
.
two
i n dep en den t even ts th e p esp ec tive p ro
kn o wn , to fi n d th e p ro b ab ility th a t b o th will
u
h
fi v m h
ys
f
y
h
u y y
u
h
s
v m
wy
f
y
h
ys
u y y
m
f m
h
mu
ss
u y y
u
i 66 f t h mb t h f il
f th
b t h v t s h pp
I
i
f t h mt h fi t h pp
d th
d f il
d i
b
h
d h pp
Thu
f t h mt h fi t f ils
d th
h t h t b t h v t h pp ;
i th
a
a
n
i
e
n
w
a
an d
S ppo s e t a t t h e rs t e e n t
a
a il
pp
y
an d s
in 6 wa s, a ll t es e c a se s b e in g e q a ll like l
o
s
e
t
a
t
p
p
'
'
a
a s an d
a il in
a
h
e n in a
w
a s
th e
ec o n d e e n t
6
pp
y
,
a ll t es e wa
E ac h o f t h e a
a
b e in g e q all lik el
6 c as e s
y
’
'
’
b e a s s o c iat e d wit e a c h o f t h e a
6 c as es, t o o r (a
b ) (a
b )
ik
all
l
el
t
o o cc r
co
o n d ca e
all e
p
q
.
’
.
n
n
da
’
a
o
’
es e
o
o
e
W
rs
(
b
(
b
en s
a
en
n
,
o
e
e sec o n
an
e sec o n
an
e c
s
)
fz
en s
a
h
an c e
t
lz
'
b
a +
' is t h e
c
h
an c e
t
)
h
at
h
e
o
a
c
a +
z
a n ce
is t h e
f
t
a
a
a
o
a
s, an
a
n
a
,
’
s
.
I
z
a
en
e
rs
e
e
e
o
’
b ot
at
h v
e
en
s
en
ts
fi th
th e
rs
a
en
a
f
a il
3
d th e
n s an
e
pp
s e co n
d
s ec o n
d
f ils
a
a
(
’
b)
a
h
(
b
a
c h an c e
is t h e
)
t
h
fi
th e
at
f
rs t
a ils
an
d th e
h pp
sp t iv h s f t w i d p d t v t s
Thus if t h
il
h
h
b
t
l
i
i
m
i
l
h
h
t
h
w
t
t
S
d
pp
pp
p
p
m
f
b
i
u
i g w ill pply i t h
f
d
d
t
y
p
v ts H
it i
yt
t h t if p
t
h
p
p
b
i
il
m
i
v
h
h
u
f
d
d
v
s
w
l
t
t
t
t
p
p
w
il
l
y
h
h
h
h
t
t
h
y
l
l
i
t
t
t
l
h
pp
pp
p
i
l
l
h
h
w
t
h
h
fi
w
h
t
t
t
t
d
h
p
p
p
p
p
il ly f
f il i p p ( 1 p ) ( 1 p )
d im
h
t
p
y
a
e re
’
an
e
,
en
se
l
e
c
a ra e
t icu la r
l
t
a
e
o
t a
—
c a se
o
an
s ee
e
e
n
ar
s
an
en
e
en
o r an
en
a re
a,
e
en
e
a
a
ar
n
en
e
rs
o
g,
en
a re
.
er o
n
en
e
’
s
,,
a
en
en
en
a
er o
n
4
e
n
a
an c e
c
o
o
o
a
3
g
c a se
e
eas
,
an c e
a
s
an c e
—
s
a
en
e c
3
z
e c
n
an c e s
a
.
a n ce
c
en c e
.
ec
re s
ec
a
re a so n n
e
en s
en
a
t
an
o
e
er
s
re s t
ar
.
h
h
v
h
a
4 57
I f p is t h e c a n c e t a t an e e n t will
i
n
en
pp
i
n
an
ill
a
n
a s s i n e d su c
o n e t ria l, t h e c an c e t a t it
e
y
pp
g
"
is p
t h e p re c e din g a rtic le
c es s io n o f r t rial
t is o llo s ro
b y s pp o s in g
.
h
u
h
s
h
p1
h
hu
w h
h f w f m
p2
p
p3
m
.
v
v
To fi n d t h e c an c e t a t s o e o n e at le as t o f t h e e e n t s
a
w
e
n
ro c e e d t
e
s : t h e c an c e
t
a t all t h e e e n t s
pp
p
is ( 1 — p , ) ( 1 — p 2 ) ( 1 —p 3 )
i
an d e x c e t in
t
s c as e s o
e
p
o f th e e en ts
st h a
i
re d c an c e is
n
e n c e t h e re
e
pp
3
q
h
v
mu
1
—
h
h
h
u
h
h
p l)
1
(
— 9
2
)
1
(
—
p 3)
m
H IGH E R
380
e
v
ts
en
h
c an
u
Thus
co n c
a
pp
'
h
is
er
aa
’
d t h e p ro b a b ilit
an
,
.
(
(
b)
a
’
b
a
’
ft
o
h
y
v
y h
u
fi
v
w f ll w
rs t
o
e
o
en
,
t,
th e
’
d p th e
ro b ab ilit
p
an
y
.
h d wh
h h
d
h d h k
u
D
h
h h
k
b
h
u
d
d
d
w
w h
22
h
h h wh
h d u
u
d
w
wh h
h l
h
h
u d x 1
igh t
f ll w
O w m
um w
wh h
u
d
d
m
u
um
h
mu
m
um w wh h
u
d
h
h
h
md m
w d w
E m
wh
k
f
h h
d w w v wh
d
k
m d w
w
wh
m d w i
w
h
h
wh
Wh
mv d
wh
hv
d w
wh
k
h
d
m d w
w
k m d w
w
h
h
k
d
E
xa
ol s
1
th e
e
bot
en o
In
.
in g an d q
te th e p lay er
t
Th e
.
?
en
51
T
an c e
c
een
fo r th e q
’
erefo re
r
th e
th e
en
c
an ce
in 52
ealt
be
c an
be
in 51
ealt
13
c
an c e re q
—
52
i re
as
re a s o n
e
t
t
s p ec ifi e
at a
ifferen t
12
12
1
51
7
13
ay s,
e c an also
ay s ,
—
o
o
A h as th e
at
h e h as th e kin g,
en
ee n c an
an ce
c
play er
.
y A ; t
at ,
t
is t fi n d th e
o f tru ID p S
at
an
a
; fo r t is p artic lar c ar
to A
e ir
)
if p is t h e p ro b ab ilit o f t h e
il
l
t
at t h e s e c o n d
c o n t in
ro b a b ilit
en t
g
p
’
o f t h e c o n c rre n c e o f t h e t w o e e n t s is p p
m
pl
y
aa
is
rre n c e
t o ge t
en
GE B R A
AL
ic
of
ol
clearly
in g is
o
f
h ic
th e q
fall t o A
'
fall
een
is
.
7
s
Th e n
b er o f ay s in
ic th e kin g an th e q een can b e ealt to A is
e q al to th e n
b e r o f p er
tatio n s o f 1 3 t in gs 2 at a t i e , o r 1 3 1 2
ic th e kin g an d q een c an b e
A n d s i ilarly th e to tal n
b er o f ay s in
e alt is 52
51
.
.
T
erefo re
13
th e
c
an c e
1
12
51
52
b efo re
as
17
.
ll
s
r
f
ro
l
2
T
i
n
s
e
f
3
b
a
a
e
a
e
b
e
o
r
a
ac
o
a
ag c o n
,
g
,
p
tain in g 5
ite an d 8 b lac
b alls , th e b a lls n o t b e in g rep la c ed b e o re t h e
s ec o n d tri a l : fi n d th e c an c e t at th e fi rs t
ill gi e 3
ite an d
ra in g
th e s ec o n
3 b lac b alls
xa
.
.
A t th e fi rs t trial, 3 b alls
an
d3
ite b alls
ere fo re
t
th e
ite
ra
f3
an
ite b all s
d 8 b lac b alls
t
erefo re at
th e
an
d 3 b lac
b alls
t
erefo re
th e
c
an c e o
03
be
ra
ra
f 3 b lac
at
th e
ay s
5 4
13
12
11
5
143
an
n
ra
ay s
3
03
;
be
ay
80
in
n
ay s
fi rs t trial
b een
a e
13
in
n
5
n
at
ra
trial 3 b alls
s eco n
ay
n
ite
3
en
2
an c e o
c
be
ay
ay b e
d
re
n
e
o
in
10
,
03
th e b ag
'
co n
tain
ay s
;
trial
s ec o n
7
.
15
t
h
ere fo re
th e
c
h
an c e o
f th e
co
mp u
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5
143
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4 29
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PR OB A B IL IT
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4 59
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'
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GH E R
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m lly l i t h t i wh
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t
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m20 ti k t m k d
c
wh
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t at it is
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ltiple o f 3 o r o f 7
is
mu
dm d h h
i
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fi
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F
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k
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5
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mh b mu
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461
It
h ld b b v d h h di i i b tw
im
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c
ltip le
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t
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it i s
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are n o
at
th e
th e
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an c e
t
f3
or
a
th e
at
b er is
n
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u
5 is
an
20
tu a lly
d th e
c
an ce
ex c lu s ive ,
at
t
en c e
th e
d th e c h a n c e t h a t t h e n u b er i s
b e e n in c o rre c t t o re as o n as fo llo s
n
e
b er is
n
f 5 is
o
ltip le
a
4
20
4
—
20
)
lti p le b o th o f 3
t ally e xc l s i e
t
e
even ts a re
ltiple
a
f 3 is
2
h ad b een
o
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20
5, it
a
ltiple
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0
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ltip le
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f 7 is
B u t if th e q
u ltip le o
3 or o
c
th e
6
re qu 1 re d c
ec a
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an
co
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Fi
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A L E BR A
h
f
df m
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car
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.
ro
t, 2
s
di
an
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or
n ar
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t wo
at n o
ac
p
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k
c ar s
fin d
are o f
:
.
n
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dra
p e rs o n s
r
t
1
( )
an c e
u l v lu
8
u
GH E R
d
in fi ve t rials
th e
c
h
an c e o
f th ro
wi
n
g
wi h
s ix
die
a s in gle
t
leas t
at
o n ce
.
d h
h
kw
fv
v wd h
v wh
vu
u iv ly
ill b e a o u rab ly re ie e b y t ree
Th e Od s t at a b o o
at is
in ep en en t c rit ic s are 5 t o 2 , 4 t o 3 , an d 3 t o 4 res p e c t i ely ;
a o rity will b e fa o rab le ?
th e p ro b ab ility t at o f t h e th ree re ie s a
9
d
d
.
v w mj
d b l k b ll
wh
d wh
ac
it e an 3
a s , an d 4 are s c c e s s
e
A b ag c o n tain s 5
at is th e c h an ce t at t ey are alte rn ate ly
dra n o u t an d n o t replace ;
feren t c o lo rs ?
o f if
10
w
d
.
u
du
o
s
h
h
h w
I n t re e t ro
11
b le ts at leas t o n c e
.
.
16
.
1S
6 25
13
h
p
a
k
ta
e rs
o e n
d
f ic e, fi n d t h e
air o
a t ran
en
th e las t digit
at
an ce
c
h
In
.
a
c o in s ,
10
re
e
a
u
r
s
p
an c e o
ft
h
ro win g
d m ml ipli d
i th p du
i
a re
o
n
t
ct
u
ro
e
h illi g
ll h illi g
h l
h f m
all s
t en c o in s a
;
t e o r er
ro
rs e an d p u t in t o t
ta en ro
t e at e r an d p u t in to t
rs t
e s o e re
s st
n th e
n
p rs e
w
h
h
t o ge t e r
l, 3, 7, o r 9
e
s
°
v ig i
h
f mh f m pu
k f mh l t
v ig i ill i
th
e re
so
c
.
wh l umb
h th
If 4
e w t at th e
12
wit
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n
n an o
I f t wo
14
ill b e 5 ea s
h d
.
15 I f 8
o n e w111 t u rn
t
e r are
fi
c o in s are
an
u
to s s e
d 5 t ails ?
c o in s
to ss e
are
d
s
s
s
n
.
atter, an
e
e
er :
or
whi h i
Ni
i
k
i
i
d h
fi d th
h
ht
ex c e p t
n e co
n s are
en
n e co
t
n
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c
one
ta
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an c e
t
a
th at t
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m
5 ti
d wh
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wh
at
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c
es ,
at
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c
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one
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ly
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p
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A B C i
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16
g
p
p
di i h t h fi wh u p d h ll wi p i fi d
h i p iv h
A
d B d w f m
17
i
u
v
i
i
d
g
p
h illi g fi d h i p iv h
d wi g v ig
ffi
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A p ty f p
i
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18
w p ifi d i d ividu l i i g
h h
A i
by
f6 h
19
df
idd
b
di
f tw j k y B
d 0
h B id A i wh i h
I i
t
ll h h
l
l
i
u
k ly wi if 0 id A h i h
l
q
y
i
b l d wh
h
dd g i h i wi i g ?
v g v li v y i w k d fi d h h
20 I f
h t f v l p t d t l will iv f ly
,
,
co n
t
.
cu
t,
t
e r res
on
s :
e co n s
.
t
o s
ec
o
a
t
t re
t
o
n
o c
s
at are
o
5
rs t
o
e ra
es s e s e x
c
ts
a s
e c
e n
re
tt
t
t
ac n
a
e s
ta
n
te r
a
e
eac
n
r ze :
n a
a
.
a
e
e o
e
1
s a
at a ro
4
to
g 3
ra
n e
a
er
e as t
so
ere
n
a
ns
so
an
ere
n,
.
n
a
e ac
o
e,
t
10
s
arr
t
n
e o
s a
a n st
er
.
t e re
o r a ra c e , an
t s 2 o 1 t at
to
e
n ;
a n st
nn n
s
ess e
ec e
ac e
n
rs t
an ces o
n ex t
n
co n
rs e
ec t
an
t
c ar s , re
o
a
o rs e s e n
o rs es are
o n an a
ac
ers o n s s
a s s
e
a
ro
n o
n
n
t
.
e r re s
ar
e
e
ra
ra
e
at o u
t
an c es
en
o
.
t
t
s on e o
,
c as e a
s
o r er c u
e c
n
e
.
on e
n
an
n
s
ea
on
ect
.
4
u
,
rec
to
s
r
es
r
e
e r
es
,
n
,
e sa e
,
.
t
en
n
c
s c
an ce
e c
an c e
PR OB A B I LI T
Y
38 5
.
462
Th e p ro b a bili ty of th e h app en i n g of a n even t in o n e
tri a l b ein g kn o wn , requ i red th e p ro b a b ility of its h app en in g o n ce,
ex a c tl
11 tri a ls
i
n
twic e, th ree ti es ,
y
.
m
.
y
h
v
y h
L e t p b e t h e p ro b a b ilit o f t h e
a
n in
o f t h e e e n t in
e
pp
g
a s in gle t ria l, a n d le t q
l p
t e n t h e p ro b a b ilit
t at t h e
t ll
ill app e n e x a c t l r t i e s in n t rials is t h e ( r l ) t e r
e en t
in t h e e x pan s io n o f ( q
v
w h
y
Fo r if we
h
m
:
ul
v
m
f
r t ria ls o u t o f t h e t o t a l
y p
n
b e r n , t h e c a n c e t a t t h e e e n t will a pp e n in e e r o n e o f
' ""
A
rt
th es e r t ria ls a n d a il in a ll t h e re s t is p g
an d a s
[
"
a s e t o f r t ria ls c a n
a s , a ll o f
ic
b e s e le c t e d in
a re
0
i
e
a ll
a
l
i
re d c a n c e is
c ab le t o t h e c a s e in
o in t , t h e re
q
pp
q
p
s e le c t
um
h
an
a rt ic
h
f
ar set o
u y
h
wy
7
Cp q
n
r
I f we
ex
p
d (p
an
g)
.
u
—r
.
r
b y t h e B in o
”
v y
wh h
h
mi l Th
a
e o re
mw h v
e
,
a
e
”
p +
m f th i
b
b
ili
t
i
f
h
t
h
p
pp
t im
2 t im
i
t
hus
ro
th e
te r
s
es o
a
wi
h
4 63
es,
I f th e
.
(
c e,
c
t
an c e
h
)
th e
“
e
a
v
en
m
es,
pp
en s
n
pp
it app e n
a t lea s t r
Of
fi
failin g to t
are
t
ro
.
is th e
su
hu
T
H
gle t
ro
s
.
o
r
h
c
H
A
.
n
ti
y
ex a c tl
t
or
es,
es
n
ti
an c e
m
u h w wh
a
p air o
rs t
c
b le ts is
n
it
es ,
t
an ce o
5
ree
f
du
o
ree
ter
(1
4
es, o r
th e
e
1
es, n
re q
—
ire
ic e ; t
6
n
6
,
e
en
at
ex
o
f
;
p
an
o
f
an c e o
f
an s io n
is th e
c
h
h
w du
u dh
d th e
t fo llo
ere fo re
th e
r
-
1
6
o n c e,
on
e re o re
f th e
ic e,
36
s
s
d wh
e e x p an s 1 0 n
s o
5
f
o
u d v
w h
t
m
m f th
ti
s
b let s is
N o w th e
-
6
v ly
m
ti
n
a
r a s
n
l ter
th e
c ti
e
p
o re
es o r
s r
+
f th e fi rs t t
th e
.
ro
o
fo
n
en
—r
th e
h w h
h wd u
h w u m h
h
m
a s in
In
v
re s
m
f il ly
m th f
t im
t i l i
t im i
en s
a
In fo r t ro s
E x a p le 1
t ro in g o b lets twice at le as t ?
h w du
e
t
.
h
h
t
in g o f t h e
t ria ls
p
re se n
r
(p
m
en
re
+
m
su
h
it
at
n
ti
r
n
p
o r
a
w ill
s e rie s
s
e
es, n
t
o
s
c
if
th e re q ire
4
5
1
(6 6)
an c e o
f
o
b le ts
c
an c e
38 6
GH E R
HI
m
AL
GE B R A
.
m wh h
um
d h h d w
d
w wh
hv
w
m k
dw h v
whi
ic are
b er o f b alls , s o e o f
A b ag c o n tain s a c ertain n
E x a p le 2
ra n an d rep lac e ;
ite ; 8 b all is ra wn an d rep lac e , an o t er is t en
ite b all in a s in gle trial, fin d
an d s o o n : ifp i s th e c an ce o f dra in g a
a e b e e n dra n in n trials
o s t li ely t o
ite b alls t at is
b er o f
th e n
wh
fin
d
h
um wh
h
h
d w
wh v u
h
d
.
.
Th e c
fo r
f
al e
an ce o
at
in g e x ac tly r
te b alls i s
o f r t is exp re s s io n is greate s t
ra
N OW
u
s
cc e s se s an
_1 p
r
q
.
—( r
n
to
II
-
,
h
uh h
c
d qn
t at p n is
fail res
u pp
S
th e
en c e
u
( 71 + 1 ) P> (p
q) r
u dv u
f
re q
an
ire
al
.
e o
is th e greates t in te ger in
r
m t lik ly
in teger, th e
os
c as e
e
h
is t
at
o
f pa
.
h
h
h
s
r on
e
p
o se
5
t
at
t
h
e re a re n
u y
m
2
t ic k e t is £
t ic k e t s in
y
lo t t e r fo r a p riz e
t ic k e t is e q a ll lik el t o w in t h e p riz e , a n d
e n s in c e e a c
w h o p o s s e s s e d all t h e t ic k et s
u s t win , t h e
on e
al e o f
.
each
r—1
a e
(n
is
n
4 64
o f £x
t
a
p q
n
T
Bu t p + q = 1 ;
If
—r
n
lo n g as
so
s
r
at
e
an
.
i
n
;
o
h w
t
er
b
o rds
t h is
a
y
m yv u
w uld b f i mt
t
i
k
h
d
t
m
i
t
p
g
o
a r su
a
e
o
w
h
h ti k t ; h
p
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b
b ly
i
t
h
i
t
b
d
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h
t
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k
t
£
p
p
p
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b
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;
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th w t h
It i
i t th
t i
£
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v
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£
t d
t h f ll w i g d fi iti
If p
t
h
u
u
t
f
i
p
p
y
d M th
m f m y wh i h h w ill iv i
f u
h
m f m y d t d b y p M i ll d h i p t ti
f
o
r
p y
a
c
eac
re a s o n a
ex
on e
an
x
t
o
o
o
u
s
r ce
e rs o n s
c
v
or
s,
e
co n
en
en
c c es s
n
y t
en
h
at
ca
e
c
e s
s
c
is
u se
th e p
h
ven
an
c as e o
n
u
u
ec a
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ra s e
p
in
ro
u
h
v
h
k m
d h
d
d
m
v u
h u
h h
v
u
u
h
h
mv d w
mv d
d h
on
f
b a b le
a
.
th e
c
c
t at th e
t at it h as
an c e
an c es
so
o
is in th e fi rs t p rs e is
t ice an t at it h as n o t
ere ign
e
eq
o
al
e
re
.
re e re n c e
.
Th e
n
c c e ss ,
s
.
.
a e
o
en
On e p rse c o n tain s 5 s illin gs an d 1 so ereign :
Two c o in s are ta en fro
th e fi rs t an d
p rs e c o n t ain s 6 s illin gs
t h e s e c o n ; t en 2 are tak en fro
t h e s ec o n
an d place in
fi n d th e pro b ab le al e o f th e c o n ten ts o f eac p rs e
E x a p le 1
e s
e st
o
s ex
e
exp ec ta ti o n
y
ien t l
s
re c e
s
a
o
e
e
co n
s
r
a
a
.
an c e
c
e
e
o
t
e
an c e
o s s es s e
o
on
n
mw
a
e
’
en o
sa
e rs o n
c
on e
my
m
o
s
a
o
I n th e
t o a p e rs o n , w e
a
i
l
t
hi
n
e
d
t
o
pp
g
.
t
e
s
on e
465
as
o
n
re s e n
e su
e su
e
or
e
re
an
s
e
u ce
ro
ec
o
as
en c e a
e
at all ;
388
H I
h
GH E R
v
W e s all n o w gi
466
re s lt
an d in t e re s tin
g
.
m
u s
e
AL
GE BR A
t w o p ro b le
.
m wh i
h
v
m
s
c
u
le a d t o
se
fu l
.
Bw
a n t re s p ec ti ely
Two p lay ers A an d
an d n p o in ts o f
E x a p le 1
in n in g a set o f ga e s ; t eir c an c e s o f in n in g a s in gle ga e are p an d g
n ity ; th e s t a e is t o b elo n
o f p an d q is
ere th e s u
re s p ec ti e ly ,
g to
a es u p h is s et :
eter in e th e p ro b ab ilitie s in fa o r
th e p lay er wh o fi rs t
o f e ac
play er
w
m h
m
mk
.
v wh
h
Su
w
u
k
m
d
m
vu
.
h
pp o se t
m
m—
las t ga
t is is
h
h
+r
ml
_
w
in
s
in
ex ac t ly
m+
m;
m
+
m
ga
1 o u t o f th e p rec edi n g
—1
in — 1
r
M
0 _1 p
g p’ or
p
m
d
1
0
e an
A
at
m
r
m
m
1 ga
r
q
es
s
Th e
.
r
c
h
th e
an c e o f
n
.
m
m
d
d
d
m
m m m m
h
h w
w
mu t wi
to do thi s h e
es
m
m
N o w th e s et ill n e c e s sarily b e ec i e in
ay
+ n — l ga es , an d A
or
1 ga e s ;
win h is
+ 1 ga e s ,
+n
ga es in ex a c tly
ga e s , o r
in s th e s e t b y gi in g to r th e
t ere fo re we s all o b tain th e c an c e t at A
’
’n +" l
m
"
1 in th e exp re s sio n
n
C _1 p q
T
s A s c an ce is
al e s 0, 1 , 2 ,
mm
h
v u
h
.
.
hu
.
m
1
m
(
)
m
I
+m
q+
p
1 2
h
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mil
si
m
-
v
arly
B
’
s c
h
is
an ce
w
u
m
m m
Pro b le
o f Po in t s ,
Th is q e s t io n is k n o n a s t h e
an d h a s
an
o f th e
o st e
i
n en t
at e
e n age d t h e a t t e n t io n o f
a t ic ian
g
I t wa s o rigin all p ro p o s e d t o Pa c al b y
e o f Pa s c al
s in c e t h e t i
t h e C e alie r de M ere in 1 6 54 , a n d wa s dis c s se d b y Pa s c a l a n d
s e l e s t o t h e ca s e in
n ed t e
co n
w ic h t h e
e r at , b u t t e
t
e ir re s lt s we re a ls o
l
a e rs
e re s pp o se d t o b e o f e q a l s kill
p
Th e o r
lae w e a e gi e n a re
or
e x ib it e d in a diffe re n t
a
a r fo r t h e
o rt , a s t e
e
rs t t i
a ss i n e d t o M o n t
i
e
n
a
r
o
k
p
p
g
Th e s a e res lt wa s a t e rwa rds o b
o f h is p b lis e d in 1 7 1 4
a s b
a ran
e a n d L a la c e
L
an
t a in e d in di fe re n t
d b y th e
,
g
g
y
p
was t re at e d e r
ll
n de r
lat t e r t h e p ro b le
a rio
o di
fi c at io n s
“
my
m
mh m
y
.
s
s
hv
u
F m
hy fi
hmv
h
y w u
u
h
u
h
f m
f mu
hv v
m
hy
fi m w
m u
u h
f
f
wy
m
v y fu y u
v us m
.
.
.
h
d w hf mk d m f h
m
h w
d m wh
h h
um
h dh
u
m
d
d
S
f
um w
wh h d m
f
um w
w h
um h w w h v
h m u
m
E x a p le 2
are
ex
th e
t
ro
n
ib ite
s
T
.
at ran
all
be
ere are n
o
eq
al
in c e an y o n e o f th e
n
b er o f ay s in
Als o th e n
t eir s u
is e q
h
fo r t is
1 , 2 , 3,
at
,
ice
is
it
th e
fac es
c
an c e
fro
ar e
t
at
th e
su
to p ?
fac e s
ic th e
ay
be
ic e
e x p o se
ay
on
an
y
t
arise s o u t o
k
b e ta
en
so
dm
one o
f th e
n
ic e ,
e
p fo r
fall is
hi c th e n
b ers t ro
b er o f ay s in
fi c ien t o f x p in th e e x p an s io n
al t o th e c o e f
fi cien
c o ef
1 to
; if t ese
o f th e n
b ers
w
d
f th e ifferen t ay s in
as to fo r
p b y a ditio n
.
ill
n
o
f
wh h
n
ic
o
a
f th e in
d
ice s
PR O B A B H ATY
N o w th e
ab o
hv h
We
a
t
e
v
"
x
(1 +
e e x p re s s io n
a4
1
(
No w
u xf
xf
(1
+
u
du
h
hu
s o
to ge t
b tain
—1
)
1P
h
e s e s e rie s
+ 55
x
1
(
”
x
+ 1)
n
d p ic
k
th e
e re
re q
u d
Th i
ire
in th e
x”
e x p a n s io n
f
o
)
n
e r an
f
o
l)
n
(
(
l)
u —
n —
2)
fi c ie n
c o ef
o u t th e
t
f
o
xp
" in
r
|
p
— n
(n
n
wh
t
f 1)
n
~
)
n —
(n
n
—x
1
(
)
M ltip ly t
c t ; we t
p ro
(
n
2
fi c ie n
co ef
to fi n d th e
e re fo re
389
.
s e rie s
is t o
p ro b ab ility is
u
d dvd
tin
o b tain e
co n
—2 — 1
f )
—l
)
v
lo n g a s n o n e gati e fac to rs
i i in g t is s eries b y f
e so
by
h
m
Th e
appe ar
.
v d w pub li h d b y h im
mh
u
f qu t u tility
i
L pl
f w d b t i d t h m f ml b t i mu h
li
h
i
i
m lb iu m
d
t
m
m
t
t
d
;
pp
p
i
i
m
i
th
i t
f
t
v
u
w
i
h
h
h
m
d
h
t
p
l
h
l
b
i
t
t
li
i
m
v
t
t
i
t
i
t
h
m
d
p
p
i
di
th
u d th
th
O t h i p i t th
d
Hi
u lt T dhu
my
y f P b bili y A t 9 8 7
l
i
b
e
s du e t o D e M o i
p
et o d o f
1 7 30 it ill s t ra t e s a
s
n
ro
ter
ac e a
a
o re
or o
a
s t ra t e
o
re c t o n
o
e
as
e
o
o
n
re
c o se
ro
’
t er s
o
s to r
e
an
at t e
.
c
e
a
n
,
on
e
e
e sa
e
e re a
o n
r
c
.
er
.
.
h
fA w
m
i wh
mk d
i
f
h w im wh t
A
2
l f
h
f b i i g
i h
i f v u
h f
I
fg m i i
t
f h wi
3
h
h h pl y wh wi
f h p v i u g m wh
i
h
f h
f u
h
h fi
l
g m h ll wi
b g
v ig d
Th
i
i
f wh i h
4
u k w i f qu l v lu fi d wh h y mu b if
h
f d w i
h illi g
h p b b l v lu
1
.
o
s
t
co n
e c
an c e o
t
t
e
e
re
rs t
e re
e res t are
e
ro
a
e
ta
o
s
e s
a
.
t
t
o
o se
o
e ac
n
.
o
to B
o ut of5
’
I n a c e rtain ga e A s
in n in g 3 ga es at leas t
.
n
a
ac es
n n
a
n
t
are
co n s
no
n
co n s o
ra
3 to 2
:
fi n d th e
c
an c e
a
es
t
s
t
s
c
a
12
a
s
2
,
ro
n
5
o
e
n
t
at
;
s
.
c
n
a
n
e n ext
t
a
1
o
an ce
e as t o
e
s
e
a
n
t
s
5t
a
es :
12
o
re e a t
9
s as
2, 3
ar e
to ta
are
a
s
’
.
at
e :
a
e o
s
a s et o
a
kill i
t
a
as
s
a
n
o
an
c,
n
ro
e
u
c
ec
XXXI I
.
u a,
se
.
o
E XA M PLE S
m
ca
e su n
n
or
n
e
s
.
e
e
r
ts
as
en
e sa
a
a
o r
n
e e ar
co n s
a
a n e
an n e r
s
e x s en c e
e
an e s
s o
ar
re a n
t
o
r o
o
a
e
t
er
n n er
e
n s
o
r
are
at
t
so
e
e re
ns
st
an
e
HI G H E R
390
GE B R A
AL
.
d tim wh t i h h
h h h d will
u mb f im ?
i i g
F m b g
v ig d hilli g p
6
i
i
ly ; fi d h v lu
i
ll w d
d w
i di
im
f hi
i
p
h wf
k wh i h i b w b y h
Si p
7
wh fi
h w h d wi h p y if h y h w i u i fi d
h
h
f th f u h p
mk d
l
i
d
i
w
i
d
b
h
C u t
8
g
p
d w d pl d Th p i b i g p d h im wh t
l f6 ?
h
i h
f b i i g
i wh
f
mk d d i
A
im
wh t
d
9
h
dd g i h m f h u mb h w b i g l h ?
h
f t h wi g 0
ly i
t h w wi h
1 0 Fi d h
di
Tw p l y
f
f q u l k ill A
l
i
d B
11
p y g
m h y l v f pl y i g wh A w
dB w
p i
If h
k i £ wh h ugh h k ?
d B h w wi h
d i if A h w 8 wh t i B h
A
12
f h w i g h igh
u mb
v ig d f u hilli g t ki g t
A h d i hi p k
13
w
i
d mh p mi t giv h m B d (7 Wh t i
h f0 p i ?
th w
h w wi h i gl di wh t i h h
I fi
f h wi g
14
ly ( ) t h
l
t
( ) h
h i i gl h w with
b t wi h B f
1 5 A mk
w di
h will t h w
v b f B h w fu E hh p i
ul u ly u il f h mwi fi d B
f di
d h y t h w im
i
p
mm ub
h
d th
h w tw d i
h
A p
16
i
i
h
f
b
u
h
d
u
w
k
l
h
l
h
m
b
g
g
h d ; wh i h h
h th m f h
f h
u mb h w i l h ?
A b g
i
i
d
u mb f h i
f v l u 11
17
wh gg g v lu i m A p
d w
im ill h
d w h i M fi d th lu f hi p t i
If 6
i k t num
b
d
d
18
l
i
b g
p
d h
d w
hw h h h
h h m f th
u mb
h mi qu l 6 i
A c o in is to s s e
re s en t its e lf an o dd n
p
5
n
.
ro
.
s
o
a
to
e
e c ta t o n
t
rs t
t
t
; t
e s ta
s a
e rs o n
e o
s ex
t
on
e one
c c e ss o n
s
n
a
an
,
t
eat e
re
n
,
t
s
one
t
ree
5 s to s s e
t ro n e
e rs
e x ac t
1
4 t
es ,
a
es :
a
t
es s
n
15
an
t
ro
one
n
3
s
eac
t
ce :
ro
t
s
e re
s es
ro
an
n
o
a s n
t
e
e
o n
ts
s et
an
an
o
ts 2
.
e
s
a
,
o
r s
n
e
to
an
t
e
o
se
tan eo
s
eas
s
t
’
s
an c e
s c
s
a
;
ou
n
a
.
t
an c e o
e c
s
ro
n
.
53 t o 2 3 t at n a s
t ro s o r
o re
on e o
t e
s
n t
n
.
.
e
en
a
e
re e ac es at
2
,
t
co n
at e
re
ro
ta
s
t
e
ro
ac
.
ce, o n e
o
n
ree a re
t
e
s
c
e s
ra
e
an
n
a r
as a
’
s
n
ns :
s e
t,
s
a
to
t
a
e o
e
e
1
,
t
e o
ta
e n
at
e
en
n
o
su
s
1) (6n
s
one
ec at o n
6n
e
2)
c
er o
a n
an
ra
3n
( 6n
e, an
c
ac e
an c e
c
s ex
0, 1 , 2 ,
t at t
n
es t
o
e
e rs o n
e
on
e co
t
er
t
t
e
e
5
.
e re
ou
s
o
e va
n
t
at
t
t
er o n
n s a co n
a
e co n
ers o n
ts 3
t o ta
a
n
.
a
t
,
a
ec t at o n
ro
e rs o n
a
are
an
t
are o
e
ro
e
an
en
et a s o
oc
ro
.
t
.
n
e
ro
a
n
an
,
n
t et ra e ro n , t e n
o
t e t e t ra e ro n
n
ers t ro
s n o t ess t
t
3
n
s
ar
o se
an
n
ea
er
o
es a
a
e
a
t
s
t
ve
.
ra
t
e
e n
e n
ro
a
ro
s ex
.
n
e
o
t
o
at s
’
n
e c t at o n
c as e
t
a
e
to
s
ac e
ar e
o
e
o
o
n
o
ce
a re
c
t
are
t o ta
e su
er n
at ran
ce an
ex
s
n
e,
e rat o n
a
an c e
ree ac es ex a ct
o
o
e
t
a
.
t
t
ers
a
o rt
t
3
t
at
.
ac es are
16,
s
n
.
1
t
n at e
s cr
l , 2, 3
n n
c
e
e rs o n
e o
a n st
an
ro
e
an c e
n s an
ere
en n
a
e
o se
ea
e
co n s
o
ta
a
e
.
t
o
o
.
o
e c
.
es
t
.
t
n
.
ga
ac e
s a
e o
ce
t
s
so
o r a s ta
rt
ar
co n
.
3
2
n
t
o
e rs
an ce o
.
t
es
n n
ro
ea
e
re
an
e c
a re
s
n
o
n
t
s
ro
an ce o
.
ra
ta
co n s
t
ers o n s
x
e c
2
t
er o
co n
ra
a
.
.
o
a
a
es ,
an c e
at
a
o
t
t
er c o n s
e
t
e
.
1 are
t at t
ac e
e su
n a
o
a
,
e
392
HI
ms
ti
fi
e
H
.
u
p
rs t
seco n
en c e
t h e p ro b ab ilit
5
is
rs e
GH E R
11
an
’
d th e
AL
y h
p
t
ro
GE B R A
at
th e
b a b ilit
.
so
y th
v
e re i
n
g
it
at
m f mth
m f m th
ca
ca
ro
e
ro
e
e
e
d is
m
h
u
u
u
um
h
m
hu
u
huh
h w w h
f y y mm
m
m
h
h w
y
w
m
u
s
h
um
u y
um
m
m
um
h w w
y
m
h
why f h u
f
h
h h
um
m h
hv
w
m y u
i
i
l
f
l
t
u
h
Th
b v i t
m
p
g
wh i h i d t J m B u lli d w fi t giv i th A
igh t y
ft t h u t h s
17 13
di p u b li h d i
C j
h B u lli t h mmy b u i t d f ll w
d
469
I t is i p o rt an t t a t t h e s t de n t s a t t e n t io n s h o ld b e
t
i
o n t a t h as b e en
dire c t e d t o t h e n a t re o f t h e a ss
a de in
p
t h e p re c e din g a rt ic le
T s , t o t ak e a p a rt ic la r in s t a n c e ,
i
t
a
r ec t l
a lt o
ro
s
e
s
6
0
t
e t ric a l die it
i
n
a
p
g
y
ro
n
t
e x ac t l
1
0
i
a t a c e is
n o t ha
t
n
t
e s , y e t it
e
i
l
l
pp
do b tle s b e a t o n c e a d itt e d t a t if t h e n
b e r o f t h ro ws is
c o n t in
b e r o f a c es t o t h e
a ll
in c re a se d t h e ra t io o f t h e n
ill t en d
n
o re a n d
o re n e a rl
b e r o f t ro s
t o t h e li it
1
ace s o
ld a pp e a r o t e n e r t a n
on e
T e re is n o re a s o n
6
b e r o f t i e s t at e a c h o f
a n o t er
e n c e in t h e lo n g ru n t h e n
ill b e a pp ro x i a t el e q al
t h e s ix fa c e s will a e a p p e a re d
*
.
’
.
.
e a
on
e at
e
ec ta n
n s an c e
o
ue
s
c
o
e
s
,
e rn o
.
es
a
’
s
a rt c
s a
e rn o
,
n
,
eo re
a
ar
c ase
e a rs
e
e en
en e ra
rs
as
an
a
o
en
a
n
er
as
n c a e
e o re
e
e
o
’
or
a
o
rs
s :
If p i s th e p ro b a b i li ty th a t a n even t h app en s i n a s in gle tria l,
s a
b er of tria ls is in de n i tely i n c reased, i t b eco
th en if th e n u
i t of th e ra tio of th e n u b er of su c c es s es to th e
c erta in ty th a t th e li
n u
b er of tria ls is equ a l to p ; in o th er wo rds , if th e n u b er of
N
a y b e ta ken to b e
tri a ls i s N , th e n u b er of s u c cesses
p
m
m
hu
u s h
fi
m
m
m
e
m
’
Se e To d n t e r s H i s to ry
lli t e o re
is gi
o f B e rn o
E n cy c lop cedia B rita n n ica
m
’
m
m
.
P
i
l
i
t
C
r
o
b
a
b
f
y
o
v
,
en
in t h e
h
a rt ic le
f
A p ro o
Pro b a b ili ty in t h e
a
t
e
r
p
VI I
.
.
m
A n o b s erved even t h as h app en ed th ro u gh s o e On e of a
47 0
n u
b er of
u tu a lly ex c lu s ive ca u s es
requ ired to fi n d th e
r
o
p
b a b i lity of a n y as sign ed ca u s e b ei n g th e tru e o n e
*
m
.
m
,
.
u
Le t t h e re b e n c a s e s , a n d b efo re th e even t to o k p la ce s u pp o se
t at t h e p ro b a b ilit o f t h e e x is t e n c e o f t e s e c a s e s wa s e s t i at e d
at P
P2 , P3 ,
P”
L e t P de n o t e t h e p ro b ab ilit t at w e n t h e
“
th
r
c a s e e x is t s t h e e e n t will o llo
after th e even t h as o c cu rred
t“
it is re q ire d t o fi n d t h e p ro b ab ilit t a t th e r c a se wa s t h e
h
y
‘
ue
tr
u
.
u
one
.
v
-
'
h
r
f w
y h
u
y h
u
h
m
PR O B A B I L IT
v y
h
um
Y
393
.
h
fi
u
h um
v f w
wh
v
h
h
wh
v f w
b e r N o f t ria ls t e n t h e rs t c a s e
C o n s ide r a e r gre a t n
b e r t h e e e n t o llo s
e x is t s in P , N o f t e se , a n d o u t o f t is n
’
ic h t h e e e n t
ila rl t e re a re p 2 P2 N t rials in
in p , l , N
si
s
o llo
ro
t h e s e c o n d c a se ; a n d s o o n fo r e a c
o f th e o t er
ic h t h e e e n t o llo s is
c a ses
H en c e t h e n
b e r o f t ria ls in
m
m
f w f
u
.
P
(p , 1
+
um
h
u
y h
u
um
P
p2 2
+
+
n
wh h
v
d th e n
b e r in
ic
e n c e after t h e e
p PN
w a s t h e t r e o n e is
an
r
t
h
ca
r
u
th e
y th
n
v
)
en
N
o r
,
N 2 (p P) ;
t wa s du e t o
t t h e p ro b a b ilit
en
th e
at
y th
th e
at
u
m
r
ca
th e
se
ca
r
u
is
se
N 2 ( p P)
.
e
v
en
t w a s p ro d
u
ce
d b y th e
r
P
p
is
se
e
P
N
pM
2
is , t h e p ro b a b ilit
at
P
p
y
h
u
y
w
47 1
I t is n e c e s s a r to dis t in g is c le a rl b e t ee n t h e p ro
h a b ilit
o f t h e e x is t e n c e o f t h e s e e ra l c a s e s e s t i
a t e d b e o re
f
t h e e ve n t , a n d t h e p ro b a b ilit after th e even t h a s h app en ed o f a n y
t
h
a s s i n e d c a s e b e in
e tr e o n e
T
h
e
r
r a re
a ll
a lle d
o
e
s
c
g
g
a p r io ri p ro b a b ilit ie s a n d a re re p re s e n t e d b
P
P
P
P
,
,
n ;
y
t h e la t t e r a re c a lle d a p o s te ri o ri p ro b ab ilit ie s , a ii d if w e
e n o te
w
a e
ro
t e
by Q
e
ed t at
Q Q
Q
p
*
.
y
u
u
v
y
u
m
f m
.
uu y
.
hm
hv
Q
wh
2
r
v
1
(p
h
)
hy p h i
u
f th
F mth i
u lt i pp t h t 2 ( Q ) 1 wh i h i h
ly
d
w i v id t t h v t h h pp d f m
f h
u
f
h
m
f
h
f
h
W h ll w giv
th
p
p
i l wh i h d
t h p i ipl
u itd
d p d
di g
a b ilit
r
d
n
o
t
s
t
h
e
o
b
e
e
p
p,
“
c a se
e x is t e n c e o f t h e r
e re
o
e
o
t
s
e
se
ses
e ca
e
f th e
e
v
en
t
th e
o n
t
o
es s
in A rt
.
a
469
e
e a rs
a
en
as
=
a
a
en e
o n e an
ro
n o
e
o es
c
er
an o
n o
t
e
ro o
en
o
on
t
e
e
t
e o re
r n c
m
.
o
t
on
er
one
t
o
e en
e
re
n c a e
.
47 2
A n o b s erved even t ha s h app en ed th ro u gh
n u
b er of
u tu a ll
e
x c lu s ive ca u ses
u
i
r
d
t
o
r
e
e
y
q
b a b i li ty of a n y a ss ign ed c a u s e b ein g th e tr u e o n e
*
s
c
,
.
a rt c e
n
t
re s
as
en
e s
ce
o
.
ro
,
y
)
d
m
so
fi
m
e o ne
qf a
th e p ro
nd
.
h
u
u
h
L e t t e re b e n c a s e s , an d b efo re th e even t to o ls p la ce s pp o s e t a t
'
t h e p ro b a b ilit o f t h e e x is t e n c e o f t e s e c a s e s wa s e s t i a t e d a t
P1 , P2 , P ,
en th e
P
L e t P de n o t e t h e p ro b ab ilit t a t
1
"
c a s e e x is t s t h e e e n t
r
ill o llo
h
a
n tec eden t p ro b a
n
t
t
e
e
;
“
b ility t a t t h e e e n t wo ld o llo
ro
t h e r c a s e is p P
u
y
n
h
v
h
u
v w f w h
u f wf m
.
m
y h wh
u
r
.
394
HI
GH E R
AL
GEBR A
.
yh
u
ih
at t h e r
t
a s e wa s t h e
ro b a b ilit
c
r
i
r
i
s
h
o
t
e
o
a
b
e
t
e
Q,
p
p
ih
t ru e o n e ; t e n t h e p ro b a b ilit t at t h e r c a s e wa s t h e t r e o n e
is p ro p o rt io n a l t o t h e p ro b ab ilit t h a t , if in ex is te n c e , t is c a s e
o ld p ro d c e t h e e e n t
Let
h
u
wu
y h
y
v
Q
Q
2
.
u
u
h
62)
1
P
n r
”
h
s
m th
H e n c e it a pp e a rs t a t in t h e pre s e n t c la s o f p ro b le
il
a
a
r
s
t
ro d c t P
l
a
e
t
o
b
e
c
o
rr
c
t
l
i
a
t
e
d
s
t
e
e
p
pr,
ill b e o n d t a t P, , P2 , P3 ,
in
an
c a se s ,
e e r, it
o
a ll e
a l, an d t h e
li
o rk is t h e reb
ch si
e
d
q
p
u
my
u
r
m
pl
u
w hv
y s m
hwv
w
fu h
w
y mu m fi
h
h
wh
h
w
k
d w d h h
m m
b
b
u d
fi
s
e
st p ;
e
a re
.
k
ite b alls an d 2 b lac
3 b ags e ac
c o n tain in g 5
b alls , an d 2 b ags eac c o n tain in g 1 hi te b all an d 4 b lac b alls : a b lac b all
a in g b een
ra n , fi n
th e c an ce t at it c a e fro
th e fi rs t gro p
E
xa
e.
hv
T
Of th e fi ve
ere are
ags ,
elo n
3
g to th e fi rs t gro p
3
Pl
If
a
3
b all is
d mh
d u
m
s elec te
b ag is
; if fro
th e
fro
o
c
h
an ce
t
h
t
at
-
e
6
P1 P1
th e b lac
kb
h
mh d f
ti ul
u
b
ili
b
ty
p
th e
ar
p
th e
we
47 3
W
v
e n ce
u
h
an ce
is
h
hu
2
c
an c e o
t
s
f
pl
d w b k
P
3 2
ra
in g a
2
lac
3
8
25
mf m
e
ro
one o
u
f th e fi rs t gro p is
15
8
__
43
25
'
v
b s e r e d, we a re a b le b y
A rt 4 7 2 t o e s ti at e t h e p ro b ab ilit
et o
o
o f an y
ar c a s e
c
b e in g t h e t r e o n e ; w e
a
t
h
n
e s ti at e
e
y
ro a
a
n in
i
n
a
n d t ria l, o r
e
e
c
o
o f t h e e en t
pp
g
n d th e
r
a
fi
h
ro b ab ilit
f
t
h
rre n c e o f s o
o
t
e
o
o
c
c
e
e
y
p
.
en
an
e
t
en
t h a s b e en
.
v
m
e ve n
;
5
all c a
35
s ec o n
a
Pap s
6
*
c
2 to th e
an
firs t gro p th e
gro p th e
s ec o n
’
H en c e th e
5
.
2
P2
“
k
u
d h
y
m
u
h
o
m
u
y
s
m
m
.
Fo r
mpl
h
i
s t h e c h an c e t at t h e e ve n t will h a p e n
p
p
ib
“
ro
th e r
c a s e if in
e x is t e n c e , a n d t h e c an c e t at t h e r
c a s e is t h e t r e o n e is Q
e n c e o n a s e c o n d t rial t h e c h a n c e
ib
t a t t h e e e n t will h ap p e n ro
T e re o re
t h e r c a s e is p Q r
t h e c h a n c e t h at t h e e en t will h app e n
ro
o
e o n e o f th e
c a se
o n a s eco n d t ria l is 2
( p Q)
f m
u
h
u s
ex a
v
e,
u
u
,
,
v
h
f m
h
u
h
f ms m
,
.
.
,
.
h f
396
H I
GH E R
AL
GE B R A
.
hw
h y
ym
m
m
uh
y
um
w
h
u
h
hw s
wh
v
uh
m
h h
u
v
u
m
m
h
f h mu
u
ju m
w fu
f p bl m w
h ll di u s fu i h
u fu l
l
Th
i
ul
d l h u gh h
t b
i t ll t u l
d
d
g
i
i
l
l
l
i
m
t
b
w
f
u
h
h
y
fi
i
d
t
m
f
p
y p
h
W e s a ll n o w s e h o w t h e t e o r o f p ro b a b ilit
47 4
a
y
o f s tate
e n t s a t te s t e d b
b e a p plie d t o e s t i a t e t h e t r t
w
it
y
wh o se c re dib ilit
is a s s
ed to
b e kn o n
n e ss e s
W e s all
itn e s s t a t e s
a t h e b e lie e s t o b e t h e t r t
s pp o s e t a t e a c
,
w et e r h is s t a t e e n t is t h e re s lt o f o b s e r a t io n , o r de d c t io n ,
t a t an y
is t ak e o r alse o o d
so
en t ;
o r e x p eri
st
be
t e d t o e rro rs o f
il l de c e it
a tt rib
dg e n t a n d n o t t o
*
.
.
.
c a ss
e
n
ec
e
as o
th e
ro
o
s
e
e x e rc s e , a n
a
ra c t c a
an
f
v e rdic t o
co
e
t
a
s en s e
on
sc
a
t
o
o r an c e,
mm
s
ts
e re s
t
e
rn
s
o
es
s
c an n o
t
n
se
a
e re
t
a
e
ar
e
co n
r
.
h
y h
um
4 7 5 W h e n it is a s s e rt e d t a t t h e p ro b a b ilit t a t a p e rs o n
i
e an t t a t a la rge n
s p , it is
b
e r o f s ta t e
s ea s th e t r t
p
i
n e d, an d t a t
i
h
as b een ex a
i
s t h e ra t io
a de b
h
en t s
p
y
o le n
b er
ic a re t r e t o t h e
o f t o se
*
m
k
uh
m
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Tw i d p d
47 6
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v d v t i th f t t h t A d B mk t h
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PR OB A B I L I T
47 7
Y
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E x a p le
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’
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47 8
W it re s p e c t t o t h e re s lt s p ro e d in A rt 4 7 6 , it
s o ld b e n o t ic e d t a t it wa s a s s
ed th at th e s tat e
e n t c an b e
a de in t w o
a s o n l , s o t a t if a ll t h e
it n e s s e s t e ll a ls e o o ds
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H IGH E R
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b o t a s s ert t at a
feren t c o lo rs fi n d th e p ro b ab ility o f th e tr t o f th e as s ertio n
all o f if
E x a p le
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o f th e tr t
; t h e o llo in g is a c a s e
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/
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47 9
If A
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Th
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fe ren t
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p
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11
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A
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at a c e rt a n
a o
res
t
rs es , 72.
e t c an s
1
o
to
e
er o
t
an
1 000
t at
t
e
at
2n
es
e co
an c e
e ac
n
1
s
ec
s
an c e
on e
s
c
n ac c
tn e s s e s ,
as s e rt n g
e a
t
a
n
a
ee n
t e
n o t to
n
e c
ve
s are
t
es
en
s x , a re e
o
o
n
.
t
are
,
.
a e
at
2
an
ne
a
an c e
ac
a
n
t
e
,
t,
e
e
e
n
t
e
en
PR O B A B IL IT
L OC A L PR OB A B IL I T Y
Y
4 01
.
G E O M E TR I C A L M E TH O D s
.
m
y
.
u
t o q e s t io n s o f Pro
Th e a pplic a t io n o f G e o e t r
481
h a b ilit re q ire s , in ge n e ra l, t h e a id o f t h e I n t e gra l C a lc l s ;
i
an
e s t io n s
c
c an b e s o l ed b
e e r,
easy
o
t e re a re ,
q
y
*
y
u
uu
h
hwv my
u
wh h
v
El m t y G m y
F m h
w
u
h
m
mv d wh
dm
h
h
m
m d
h
h
u
h
u
P
hh d
u
uv
mv d h
k wh
h h
m
hh d
h
h d
h
m
m
u k
h
d hu
u d
E
ff
re
1
l
e
p
xa
etr
eo
ar
en
e
o
.
ro
.
o
ain
is les s t
ers
an
,
ea c
d
a t ran
.
re
o
o
e
ft
is
at
:
lin e s
th e c
al
eq
o
o
p o rtio n is c u t
t at th e s u
o f th o
f le n gt
an c e
l
a
1?
an
lac e th e lin es p aralle l t o o n e an o t e r, an d s pp o s e t at afte r c ttin g,
T en th e q e s tio n is e q i ale n t to
th e rig t an
p o rt io n s are re o e
an
at is th e c an c e t at th e s u
o f th e rig t
a s in g
p o rtio n s i s greate r
l
ea
r t at th e fi rs t s u
t an th e s u
o f th e le ft
r
n
s
I
i
s
c
t
an
t
i
o
is
o
p
s th e req ire
e q ally li e ly to b e greater o r les s t an
th e sec o n ; t
-
.
-
-
.
p ro b ab ility is
C OR
c
j
h
u
an c e
h
h h
E ac
.
t
t
at
m
o
f two lin es is
eir s u
E x a p le 2
s t a s like ly as
mi
h
re e
d
t to
n o
n o
lin
en o
es
are
te th e
to b e
n
h
h
t gre ater t
s no
If t
.
k w
c
d
si
an
I is
fa
h
u
m
w
d
h
h w
h
h
w
dm h w
h
[
]
hu
u d u
w
m
h
d w
hm
wh
dd
d hm
Of t
1
2
n o
t
e x c ee
d
in g l : th e
.
dm v h h
h
h
h
k w
hw w
h
h
k w
v h h h m
o s en
es o
h
f len gt
o
a t ran
p ro
i
l
n gle
r
i
o
s
s
b
e
t
a
p
o
e
,
t
at
t
ey
are
.
lin es o n e
o f th e o t e r
u s t b e e q al t o o r gre a ter t an e ac
o f th e o t er t o lin e s is t a t
t o ;
e n o te its le n gt
by l
T e n all e n o
th e len gt o f eac li e s b e t een 0 an d I
B u t if e ac o f t o lin e s is n o n to
b e o f ran o
le n gt b e t een 0 an d I, it is an e en c an ce t at t e ir s u
is greater t an 1
E x 1 , Co r
re e
.
.
.
T
s
th e
re q
E x a p le 3
s e
t at th e o
fo r e b y t e
t
ra
dra
.
H
.
.
ree
lt fo llo
s
.
dm
v
to a gi e n c irc le
ra n at ran
o
tan gen ts are
3 to 1 again s t th e c ircle b ein g i n scri b ed in th e trian gle
.
c irc le
A
.
res
s are
ree ra n
to th e
H
ire
T
.
D wh
w
.
d ml
o
th e
m
in e s P, Q , R , in th e s a e p la n e
s ix t an ge n ts p arallel to t e s e lin es
h
.
as
th e
c irc le , an
d
4 02
HI
GH E R
AL
G E BR A
.
md
h
vd h
w
d
d
h u wh v
u d u
w
d
mtim b
P b b ilit y my
482
Q u ti s i
id f
di t G mt y
lv d b y t h
i tly
h
h
m u d
m
m u d
dm
h
w
d
d u
b
PQ
L
mu d m w d
h mu
h
m w d
PQ m u d
h y mu
d u
Q
h
N wi
vu
u hv
w m
F AQ
Q
h
u hv
w m
T en o f th e 8 trian gle s s o fo r e it is e i en t t
to 6 a n d in s c rib e in 2 ; an d a s t is is tr e
e s c rib e
irec tio n s o f P, Q , R , th e req ire res lt fo llo s
c irc le
th e
at
ate e r
b e th e
ill b e
o rigin al
.
*
es
.
ven
:
o
c o in c i
-
n a e
e
eo
es
e r
no
e
co n
.
le n gt s a , b are
e as re
p o in t o f th e
+b+
a
.
c,
re
e as
lin es
at
ill
e.
AB
fro
re
et
ea s
co o r
o
a
e
so
a
a
On a ro d o f len gt
fi n d th e p ro b ab ility t at
E x a p le
ran
ro
n
e
so
en
on
e
’
’
'
= b,
A P z y, P
b e les s t an a c
o
fa
n
lin e ,
th e
P to
an
s
B,
s
ar
’
pp o se
’
ea s
; als o le t a b e
b +c
A ga in le t
a
z
an
.
ar s
B, t
st
en
.
rab le c as es
o
at x
t
so
A P = x an d
s t b e le s s t
'
re
fro
P to
ppo se
s
an
>
y
en ce
A gain fo r all t
e c as e s
st
e
a
+
x, o r x
p o s s ib le ,
x
a e
AP > A
’
> b +y
st
e
,
o r e ls e
A
'
>
,
.
a
e
> 0, an d < b + c
y > 0,
an
d <a+o
mk
u
u
Tak e a p air o f rec tan g lar a xes an d
a e 0 X e q al to b + c
an d C Y
,
e q al to a + 0
ra
th e lin e y = a + x rep resen te b y TM L in th e fi g re ; an d th e lin e
x
b y rep res en te b y KR
u
.
D w
d
A
P
h
T
Q
B
,
YM , KX
are eac
d
wh
d
u dh
h ul
eq a
Th e c o n itio n s ( 1 ) are o n ly
KXR ,
ile th e c o n itio n s ( 2
an gle 0 X , o y ;
th e
*
Exa
ire
req
483
l
es
p
m
“
.
c
p ar
r co
en
ro
s e ac
ar
0,
s atis fi e
GM , GT
are eac
d d
h u
eq
al
to
a.
b y p o in ts in th e trian gle s M YL an d
'
s atis fi e
b y an y Po l n ts M th in th e rec t
are
’
sh
all c lo s e
s
th i
ch a
t
e
p r with
so
m Mi
s c ellan eo
e
u
s
.
E x a p le 1
s are
to
an ce
We
m
ll
th w
tm t
h
mp tm t
ba
u
.
’
'
en
d
,
“
en
.
i
d
v
d
d
m
A b o x is
m u m m t i t whi h
h h will b p m
m m
h t i i g b b ll
i i e
n to
e q al c o
part
n at ran o
; fi n d th e pro b ab ility t at t
c o n tain in g a b all s
, q co
p art e n ts e ac
s e ac
c o n t ai n i n g c b alls an d so o n
,
w ere
d
h
.
o
,
p a + gb + rc +
h
en s
n
o
c
71.
'
ere
co n
e
a n n
oo
a
s
4 04
HI
u
I n th e p artic l ar
th e
re q
u d h
ire
c
wh
c as e
1
2
1
a
+
(
(
n
AL
GE BR A
.
k= 2,
en
n
an ce
GH E R
n
(
a
2 u + 1l
+1
+ 1)
(
2 (2 n + 1 )
3
If
fi
is in
n
d
efi n
n
itely large , th e
mt
an
h
an c e
e ’ Of
hu
dt
'
n
th e
s
c
h
an c e 1 8
t
rep lac e
'
n
Q,
—l
u
r
t
an c e
h
at
(
k)
(
n
k+1
.
'
—2
n
— k+ 1
n
(
r—
(
1)
r—
’
r
1)
r
0
th e
n
-
n ex t
d w w
ra
in g
-
n
-
(n
(
k + 2)
—1 r
)
[A rt
.
v wh
n
r
(n
ill gi
(
—k l l
)
le + 2 )
-
k+ 1
11. —
r—
—k + 2
)
(
h
en
;
— 2
r
(
c
wh
,
,
r
Th e
mit
to th e li
]
—k+ 2
)
ZPr
r=
al
1
d
r— l
r
d
u
k+ 2
are n o
(
eq
k+ 1
k+ 2
k
I f th e b alls
is
NH
1
( ii)
an
c
e a
—k
)
(
n —
r= n
ite b all :
k+ 1
2
,
z
7
°
o n
k
—k
r
)
k+ 2
+ 1)
k+ 1
k+ 2
wh h
ic
is in
m
d
’
ep en
d
en
t
o
f th e
um
b er
o
f b alls in th e b ag at fi rs t
.
dd
v
d
v
d mwh
h
v
w
d
um w
wh h
w
L
m
h
wh
h
h
w
v
h
m
u
mu h
d
h
u
m h
m
u
u
h
u
w
wh
h
i
S
)
(
h
d
d
m
um w
wh h m d
d
h wh m
d
h
d
d ( )
A p ers o n
E x a p le 3
let ters are plac e in th e
e ery letter go e s
ro n g ?
.
te th e
w
n
en
rite s 71.
elo p e s
b er o f
le tters
a t ran
d
o
en o
n
ay s
ic
a
.
at
,
p es ; if th e
p ro b ab ility t at
re ss e s 71 en
i s th e
elo
th e le tters go
ro n g, an d
rep re s en t t at arran ge
ic
let a b c d
all th e le tters are in t e ir
i f a in an y o t e r arran ge en t o c c pies th e p lac e o f an
No
o wn e n elo p e s
’
s t e it er o cc p
letter b , t is le tter
s
l
r
a s s ign e
a
r
s
e
t
e
o
a
c
e
o
o
p
y
et a n
in
en t in
an
all
.
.
’
ic
T e n th e n
b er o f ay s in
pp o se b o cc pie s a s plac e
is p lac e i s u n _2 , an d t e refo re th e
ain in g ii — 2 letters c an b e
all t h e re
ic a
ay b e
is plac e b y in t erc an ge it s o e o n e
b ers o f ay s in
n
1 letters , an th e re s t b e all is plac e is n
1 u n _2
o f th e o t er n
.
.
PR OBA B I LI TY
u
4 05
.
u
d
u d d
d
d wh h
mu
um w
wh h u
h whh
(
)
u
d
d
h
w
l
n
b
o e s n o t o cc py a s
T
d
i
e
b
s
ac
e
a
e
n
i
n
s
,
p
p
( )
c o n itio n s , s in c e a is fi x e
in b s p lac e ,
en ts s atis fy i n g th e re q ire
arran ge
ic c an b e o n e in
is p lace ,
s t b e all
ay s ;
th e le tters b , c , d,
ic a o c c p ies th e p lace o f an o t er le tter
b e r o f ay s i n
t ere fo re th e n
1 u n _1 ;
b u t n o t b y in terc an ge it t at le tte r i s n
S pp o s e
m
11
h
fro
mwh i h
Als o
c
,
u
=
1
o cc
a
b y th e
0,
u
2
’
mth
e
1; t
=
d
hu
o
=
u
<
we fi n ally
h
o
f
1
re q
ire
1
c
1
h
b tain
o
1
w
wh h
u d h
in
’
an
1
ay s
.
.
.
1
um
(
11
fA rt 4 4 4 , we fi n d
o
s
b er
N o w th e to tal n
s is n ; t erefo re th e
l
ac
e
p
1)
n
’
ic
th e
an c e is
h
t in gs
n
c an
b e p u t in
n
1
v v
mh
m ym fi
h y
m
Th e p ro b le
e re in
o l e d is o f c o n s ide ra b le in t e re s t , a n d in
a in t a in e d a
l
ac e
an
o di c a t io n s h a s
er
an en t
so
e o f it s
p
p
I t wa s
rs t dis c s s e d
in
o rk s o n
th e T eo r
o f Pro b a b ilit
b y M o n t o rt , a n d it wa s ge n e rali e d b y D e M o ivre , E le r, an d
L apla c e
m
w
m
s
y
m
fi
.
u
u
.
uj
y
v
484
Th e s b e c t o f Pro b ab ilit is s o e x t e n s i e t h a t it is
i p o s s ib le
o re t a n
a
s k et c
e re t o gi e
o f th e
rin c i a l
p
p
i
n ad
ra b le c o lle c t io n o f
a l e b ra ic a l
e t o ds
A
ro b le
s
i
l
l
s
,
p
g
’
ill b e o n d in W it o rt s
t ra t in g e e r alge b ra ic a l p ro c es s ,
Ch o i c e a n d Ch a n ce ; a n d t h e rea de r wh o is ac q a in t e d it t h e
’
a
c o n s lt Pro es s o r C ro t o n s a rt ic le Pro b a
I n t e gra l C alc l s
y
b ility in th e E n cy c lo p cedi a B ri ta n n ic a
A c o p le t e a c c o n t o f
’
n te r s
t h e o rigin a n d de e lo p e n t o f t h e s b ec t is gi e n in To d
H is to ry of th e Th eo ry of Pro b a b ili ty fro
th e ti e of Pa s ca l to
th a t of L ap la ce
*
m
.
h
v m h
m
mh
v y
w
uu m
u
f
v m
uj
h
fu
u
f
m
v
.
.
m
m
u
h w h
w h
u
hu
m
.
h y
y
Th e p ra c t ic a l a pplic a t io n s o f t h e t e o r o f Pro b ab ilit
to
co
e rc ia l t ran s a c t io n s a re b e o n d t h e s c o e o f a n
e le
e n ta r
p
a
re e r t o
h
t re a t is e ; fo r t es e w e
t
e a rt ic le s A n n u i ties a n d
y
I n s u ra n ce in t h e E n cy clop te dia B rita n n ic a
mm
y
f
m
h
m
y
.
*
1
t
it
h ww h
I
2
d w
t
.
ro
at are
t wo
u
p
di
th e
ce
?
t
h
'
E XA M PL E S
o
dd
s
n
a
rs e
e re are
wh
e
.
.
v u f h wi g l
v ig d h illi g
h
h
h h m
v i
g wi h
in fa
5so
ra n o u t o n e b y o n e ,
at is t e
a n d s h i lli n gs alte rn a te ly , b e gin n i n
.
XXXII
.
o
t
r o
e re
c
ro
n s an
an c e
t
t
at
a so
at
n
4
t
s
e
e re
y co
gn
eas
n
s
t 7 in
.
e o u
a s in gle
h
v
If t
t
so
e y a re
e re ign s
4 06
3
c
4
dra
.
s
If
h
t
In
lo ttery
a
k
t
at o u
an c e
t ic
a
v
h ip
h ip
h ti k
9
a e rage
an
on
s
s
f5s
o
all
t
e
.
f 10
out o
at
c
e ts
:
s
eas t
3
s
u
an
are
at eac
e
h
t o po rt , w
s afe
retu rn
will
d l
bl k b t
h w th
hp
s e x pe c te
d retain s it
in g th e p riz e
e t, an
G
A L E BR A
v
arri e
one
eac
;
h as
e rs o n
?
hp
an
at
ers o n
eq
ul
a
w
d b g
i
whit d d b ll d
b g
5 O
wh i d d b l F m f h m h
dm
t i
d w fi d th h
h h y f diff t l u
tw b ll
Fiv p
d md
di i th
A B 0 D E h w
6
l tiv
f wi i g
u til f h m h w
fi d h i
h
i u
ill
u pp i g h t h w
pp
b
i
u
f
h
b
d
d mwh
Th
7
g h
q
l
h t h t tw
f
d
f
h ?
i h
th
mm ub d th th
A p
t h w t w di
8
h d h umb
f big k i
th l w
gul
h d fi d th v g v lu f th h w d
f h t
h
f h wi g
mp h h
6 7
kill i
dt D
B
3 t O
A
9
fi d h
h th A i th t i l wi h h p
will u
d
wi t l
by h fi
i
b w
h
h
w
w
t k i
10 A
p
with
if h
wh
i
h
t h d l di
p
f h l
h
?
l
i
m
A
Tw pl y
A B f q u l k ill
f
11
p y g
g
d B w
w
mpl th
mp
g m
g m
hi h
f wi i g
i
v ig d w h illi g p
A pu
12
i i
d w
hh d dl k
f h mwh i h p v
v ig h w h th h i q u lly lik ly b v ig
b
hilli g
d B pl y f
A
A i
th w
di fi t
di t
13
p i
wi if h
h w If h f il B i h w d wi if h h w
I f h f il A i t
h w g i
6
d
wi w i h
d
h
h pl y
fi d h
f
d
S v
l
t
f
t
h
w
up y f h i t i
14
p
fi l ilw y mp tm fi d h h ( ) h w p ifi d
b
i
i
h y b i dj
p
( ) th
pp
h
m id
mb
i t
f 7 digit wh
v
h
15 A
mi
h
p
h
i
f i b i g d ivi ib l b y
5
Fi d th
f h wi g
h
di
i
i gl h w wi h
16
c
h
w
h
.
th e
Is
GH E R
H I
f dra
an c e o
t
one o
n
t
os n
s
t
t etra
t e
ar
e c as e o
are
co
t
ro n
t
t
an c e
eas t
ce a
ac e
an
t
an ce o
c
an ts
t
2
e r c
es
to
or
an
a
t
so
on :
n
.
e
en
e sa
e s
an c e o
o
at
,
er
e , an
e o
er
e o
ta
e n
ac e
a
e
,
t
e
en
ro
,
n
an
.
’
o
;
s as
r a s, o n e
e
3
t
t
on
co n
ete
2;
:
o
an
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e ac
’
4
s as
s
,
3
c cee
t
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rs t
e
e rs o n
4
are
t
o
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ro
at
s
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es
s
t
e
s
a
s
are
a
s e t, an
e
a set o
n
an
ts 3
t
o
a
es :
;
are
co
.
ta
3
ns
ea c
;
e
o
co
n
6
s
so
an
ns
e re
an
at
e ot
or a
r ze
t
e
t
ta
e
nu
ts
n
n
e
.
a
s,
e c
a
o
an
s
at o n e o
s
oo
er s e
n
t
a
e
s :
ers o n
a
c
,
to
e
ro
ar
o
s
en
t:
2
s eat s ,
to
to t
s
ro
a a n
eac
ra
te
s
;
s
t
o
an c e o
co
os
a
s
ers o n s
rs t c ass ra
o
,
e :
,
a
e
.
.
ra
e rs
n
ro
5
.
c
t
e rs o n s
:
to
e
an
e
e ra
e
e a so
ro
es
ere
n
s
o
.
.
n
ree
nn n
ere
n
n
oc a
to
s o n e co n
or a s
l
s
rs e
e a so
5,
es t
o
rs .
nn n
at ran
on c
e co
,
.
an o t
one o
co o
an ces o
o s en
c
o
e o r er n a
e c
ears
e n
e
e ren
n
e
e r re a
e a
n
s as
e
an c es o
.
ra
a
o ar
o s e n at ran
c
,
a re o
e
an ac e a
n
ro
s a
a
a
ro
e
as t
e
o
.
t
t
t
t
a
a s ec o n
s , an
.
c erta n
.
an
to
at
at
er o n
’
s
s s
e c
t
t
c e, o n e
;
a
re
o n e c o o u r an
ro n
an ces O
one o
t
es s
e n
e
ro
n
e
o
t
3
,
n
a c
s
,
etra
’
n
t
co n
ro
e
e c
.
to
an
an c e
s an ace :
o are o
a
.
,
,
ares o
ers o n
a re
,
s
e
e c
n
ro
ro
e
an ce
e c
.
t
t
e
ree s
.
s
n :
ers o n s
e
.
an
ra
s are
a
ta n s 5
5 re
a ls
co n
te
4
a ns
o
a
n e
.
co n
.
a
or
ro
,
rs
e
to
an
to
an
an
t ro s
6 o r 5 o r 4,
e
n
t
n
,
er
.
e o cc
t
n
a
at
e c
t
e
an c
o
1 t
an c e
o
ta
t
e s x s ea s
at
t
o
n
a
ac en t
s
59 ;
ro
n
ec
s
s e ats
e
on
.
er c o n s s s o
e n
e c
s
an c e o
11
e
t
o se su
s
ro
s
n
e
t
at
t
e
.
12
n a s n
e
t
ro
t
3
ce
.
4 08
GH E R
HI
h
AL
GE BR A
.
u
h
v
v
d
ud
do
m
h
a
T e re are t wo p rs es , o n e c o n tain in g t re e s o ereign s an
26
A co
s illin g, an d t h e o t er c o n t ain in g t re e s illin gs an d a s o e re i gn
ic ) an d ro pp e I n t o th e o t e r ;
n
n o
o n e ( it is n o t
is t a en fro
t
o b e t wo
o n
a
r
e
f
t
e
r
s
e
e ac
ra in g a c o in fro
an d t e n o n
,
p
y
s illin gs
W at are th e o s again s t t h is ap p en i n g agai n I f t wo o re
rs e ?
ra n , o n e fro
are
eac
p
h
h h
k m
k w wh h
d
h
d w
h
m h u
m
h
h
dd
h
d w
m h u
h pi k
dm
i gl i f md b y j i i
If
27
h h dd
i l p v
f
f
h
i um
i
g i
u gl d
i
b i g
i
m
f
h
f
k
d
u
m
i
Th
28
p
wh i h h
h h mf y w f h
i l
d mi d i g
h h hi d ?
d mi h p wh i h h
i d ivid d
A li
29
h h y f m h id f p ib l i gl ?
i d
igi lly
v ig d h
3 0 Of w p u
h
v ig d h lli g O p u i k b y h
d w
wh i h p v b ll v ig wh i h
i
d
h h hi p u
i
ly
v ig d wh i h p b
d w f mi ?
f h
b l v lu
d m
igh li
w p i
k
fl g h
O
31
h n h h di
h mi g
h 6
b w
fi d h
igh li
h i d ivid d i h p b y w
fl
A
32
k
d mfi d h h
h p ig
h b
i
If
igh
li
f l gh
b
w l gh
b
33
m u d
d mh h
h h mm p f h l g h
h ll
d i b wh
l
h
b
l
h
h
b
h h mll l g h b li i ly w i h i h l g i g
If
igh li
fl
h b w l gh b
34
m u d
d mh h
f h i h vi g
mm p wh i h
h ll
ddi (
d i l
i h
wh
th
b
b)
35 F
A
B
i
0
h th
D
t
g
p
g
v lli g i ilw y i whi h
i
lfi
l
m d l
d
h i d l mp m A d B g l m wh
i
i
i h
f
v lli g fi
d hi d l
p
p
di hi
by
0
d D
wh
l di
im
il
i
i h
h p
d by l m P v
p
h f ll v lu f
u
wh
h
i
l
(
p i
p
l m )
d B
m lik ly b f u d b h i h
m y f h m l dy h
h wi h diff
.
h
.
.
t
n
te
ac
e n
ne
.
t
at
t
er
an
4
o
10
e re
co n s
ra
a
at
t
t
n
s tra
.
p
ts t a
o n
a
on
at ran
re
ta
ree
t
e
t s ta
n
o
e n a t ran
3 to 1
s are
e o
o
a n st
a
t
t
e c
to
e re
t
ta
s
t
s
e
an c e
c
at
at
n s , an
t
n s , an
en
so
an ce
e c
ns :
e re
a
a rc s
e
s
e re
so
rs e
so
t
o
o
at
25
n e
e a
so
t
art s ,
n e
.
e
n s on
t
s
e
ro
e
t
t
en
t
a
et
gt
o
an c e
t
t
at n o
t
t
en
t
at
n
to t
art
t
+
a
e co
at ran
are
s
e
an c e
ta e n
re at e r t
ts
e
s
e c
n e
o n
o
een
a
t
n
;
s
ro
en
ta
e re n c e o
e c rc
an
o
co n
n
t
on
t o t ree
t r an e
na
i
o
su
n
s t an c e
e
s t ra
,
r
e
o
s
n e o
o
o
t
ro
ne o
at ran
en
.
e as
t
t
c
ra
t
t at t
ce
a
e c
e n ex t
at
or
t,
a s tra
n
.
at
o ss
15
rs e c o n
s
t
e O
a
e
t
e
a
one
an
ou
n
t
e
at ran
at ran
es o
n s
t
t
an
rs es
so
an c e
c
t
e s
o
gt
ro
en
an ce
c
e
t
t
.
e
ta
are
s
or
e
o n n
.
reate r
s
ne
t
s
e
a c rc e ,
ts
o n
at
e te r
or
s
e
an
-
re e
.
c rc e :
e
e re n c e o
e c rc
ts
t
t r an
a
.
t
s
reate r t
en
t
t
art O
s
;
.
arts
re e
o
on
an
o
an
.
are
a,
es e
o
t
en
s
9
d
a
s
C
t
no
e xc ee
0
e re c i s
s
t
es s
a or
an
a
at
t
t
e s
.
s
a
n o
er
on
a
t
t
en
es e n
s t ra
at ran
re
e as
a
o
,
e xc ee
t
t
n e
e c
t
t
re
en
o
gt
t
an c e o
a
e r
a
e re
s
e
ar e r a
+c t
+
an c e
o
s
es s
t
en
a co
n
s
e c
a
t
n
t
a so
;
an e
s
a re
a,
on
art
t
er a o r
c
.
c
our
.
tra
e
an
s
n
r
ec t ve a
re p re s e n
te
s
a
t
A
co
ar
at,
p
a ra
n
t
n
-
r or
n
c
t
an c e s
an c es
es
e sa
,
,
n
en
ts
t ra
o
,
A
e
are
an
a
co n
ta
e
eac
t
o re
’
re s e n
t
e
t
a
ea c
en
,
an
t
en
e
ere n
re
are
are
a
es
o se
o
t
o se
c as s
,
e
c as s ,
r
,
art c
to
'
t
or
te
e
e r, are
o
s ec o n
-
are
n
to
rs t c as s ,
ns
v;
re
e x ce
v
e rs
rs t, s ec o n
n
eac
an
re s ran
an
.
are
t
t
en
A, p ,
A, p ,
o
,
c
n s t an c e
c
a
,
art
e ac
n
o
t ra
co
r or
v=
e rs ,
a
c as s
or a
n
a
p
as s en
ar
.
ro
.
c as e
ot
n
one
n
e
en
n
t
e
C H A PTER XXXI I I
D E TE R M IN A N TS
v
.
.
f
u
48 5
TH E p res e n t c h ap te r is de o t e d t o a b rie dis c s s io n o f
o re e le
en ta r
det e r in an ts a n d t h eir
ro
rt ie s
e
Th e s lig t
p p
ill e n a b le a s t de n t t o a a il
in t ro d c t o r s ke t c
e re gi e n
i s e l O f t h e a d a n ta ge s o f de te r in a n t n o t a t io n in A n al t ic a l
G e o e t r , a n d in s o e o t e r p a rt s o f H ig e r M a t e a t ic s ;
a
lle r in o r a t io n o n t is b ra n c o f A n al s is
y b e Ob t a in e d
’
D r Sa l o n s L es s o n s I n tro du cto ry to th e ill o dern H igh er
ro
’
A lgeb ra , a n d M ir s Th eo ry of D eter in a n ts
m
m
.
m
y
u y
hh
v w
m
hm f
v
m h
m y
fu
f m
h
h
m
f m
486
h mg
C o n s ide r t h e t wo
.
o
mu ltiply i
n
t ra c t in g
fi
h
t
e
g
an
v
rs t
e
d di idin g b y
u
q
h
re s
ul
t is
so
mb y
2
i
we
a
t,
u
l
es
a
a
f
a
b
v
h
hm
.
s
lin e a r
e
u
q
a t io n s
Q
O;
by b
O b t a in
—
mtim w
e
z
y
y m
the
at o n
( 1,
T is
z
en e o
Qy
x +
Q
a
o
u
h
m
u
.
=
seco n
d by 6
su
b
O
.
rit t e n
l
bl
2
b2
m
h f
d t h e e x p re s s io n o n t h e le t is c a lle d a det er in an t I t c o n s is t s
it s e x p a n de d o r
eac
o f t wo ro
s an d two c o l
n s , a n d in
ter
is t h e p ro d c t o f t wo q a n t it ie s ; it is t e re o re s a id t o b e
o f t h e s eco n d o rder
an
m
w
um
u
u
.
f m h
.
Th e le t t e rs a
b u a 2 , b e a re c a lle d t h e co n s ti tu en ts o f t h e
de t e r in an t , a n d t h e t e r s a b a b a re c a lle d t h e ele en ts
m
m
m
.
10
487
f ll w t h
o
s
o
i n g th e
b
a
h
e
m
c o lu
of
mi
y
1
b1
61
a
2
b2
b2
a
4 89
l
a
m
L et
.
us
n o
w
mi
e li
n a t in
g
b
l (
x,
y
b
c
b
a
z
,
,
d
al
b1
a
b
a
b2
a
bl
2
h
e x p res s io n
O,
a x
b gy
c z
z
O,
a x
by
c z
0
f t re e ro
th ird o rder
4 90
.
ab o
v
e
w
s
an
h
3
a
e
61
01
a
?
62
0
s
b3
c
co
)
c
i—c
a
l
re a rran
n an
en
a
e
e n
o
b
c
, ( 2 3
l
(
a
b
s
0
0
0
C
2
3
3
l
b
s z)
a t io n s
O,
.
I
er
c
bl
a
m
—b
b3
u
q
.
wh i h
de t e r in a n t
c a lle d a de t e r
en
b3
n ear e
.
O
a
r
b2
on
3
n s
e
e
2
f ig
lu m i
t h e le t b
re e
2
-
a!
de ter
n sign
n s o
u s li
.
h
m
t
f
m
t
t
g
mi t my b w itt
By a
de t e r
a
d t
c a
m f th
mi t ly i
in E x 2 , A rt 1 6 ,
as
b , ( 02 a 3
1
.
s
b t a in
o
b
o n
e n eo
o
c ,z
a
d th e
o
by
c
e
h mg
th e
a x
,
we
g
at
2
c o n s ide r
ch a n
.
m
a
th e
h
t
an
3
o
m
s e en
g
an
a
t
rc h a n e two ro ws o r two c o lu
w
e
i
n
e
g
if
t, we o b ta in a deter in a n t which difers fro
n an
By
.
deter in a n t is n o t a ltered b y
d th e co lu n s in to ro ws
th e
e a s il
is ,
at
—a b
b
l ,
§, l
n s, a n
A ga in , it is
.
a
t
a
th e va lu
at
ws in to
ro
4 88
G
A L EBR A
Sin c e
.
a
it
GH E R
HI
)
s
+ d
b
(
e
c
mi
ex
p
an
c
n an
de d
c o n s is t s
t
o
f th e
412
HI
GH E R
m
Th e de t e r in an t
“
b
6
6
I ( 2 3
b i ( a z ca
h
.
01
0 11
2
3
b
a C
’
GE BR A
b1
al
b l ( c2 a 3
l
C
AL
C
)
C
l
b
a s)
C
b
a
s
I ( s
en c e
l
bl
cl
“
2
b2
0
a
b
0
a
hu
T
it
s
m
a
pp
3
t
e a rs
h
2
3
at
if
j
two
a d a c en
t
c o lu
m
f
th e
deter in a n t a re in terch a n ged, th e sign
a in s u n a ltered
c h a n ged, b u t i ts va lu e re
m
I f fo r t h e
v yw
a]
b,
01
a
b2
0
63
6
,
3
t
h
en
e
re s
m
2
3
u lt w h v j u
b
)
( l
my h w t h
th e
m
deter
de n o t e t h e de t e r in an t
e
“
by
ws , of th e
in a n t i s
.
f b re i t
s ake o
o
n s , o r ro
e
a
b t a in e d
st o
my b w
e
a
rit t e n
a e
mil
z
yw
a rl
Si
e
a
(
493
If two
iden tic a l th e deter
s
C a
ro
i
Fo r le t D b e
at
e
b
s sl
ws
(
n an
Cl/ 6
1
two
or
mi
.
s
6
2 3)
co lu
t va n is hes
v lu
(
Z
m
ns
a
6
6
1 2 3)
f
°
the
o
mi
deter
n an
t
a re
.
m
h
f t h e de t e r in a n t, t e n b y in t e r
w
c an
i
n
o
ro
r two
co l
t
s
o
n s we
b
t
a
i
n
r
i
n
an
o
a
d
e
t
e
t
g g
—D
a l e is
o se
n a lt e re d
e
n
c
e
; b u t t h e de t e r in a n t is
;
I)
D , t a t is D
T s w e a e t h e o llo in g e q a t io n s ,
0
h
wh
r
.
v u
h
th e
a
w
.
“
A
1
1
b,A l
c
A
l
l
e
o
um
m
hu h v
_
A
g
2
u
f w
A
s
3
D
b, A 2
63 A 3
0,
c
c
A
a
3
0
a
2
a
m
m
.
h
u
.
494
If ea ch c o n s titu en t i n a n y ro w, o r in a n y
u lti li ed b
h
e
a
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exp ress ed
z
2
3 ( b —c)
27
be
u a res c a n
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tities
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.
u
to E le r
ue
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-
26
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1
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f ll wi g th
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ce
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or
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E TE R M I N A N TS
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HI
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Th e le t an d
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um
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at o n
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m
h m
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w hh v
h
u
or
o f de t e r in an t is
o re ge n e ra l
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ill b e s f c ie n t h e re
b e o n d t h e s c o p e o f t h e p re s e n t wo rk it
a e b ee n e s t ab lis e d in t h e
h ic
t o re a rk t a t t h e p ro p e rt ies
c a se o f de t e r in an t s o f t h e s e c o n d a n d t ird o rde rs a re q it e
e n e ral, an d a re c a pab le o f b e in g e x t e n de d t o de t e r in a n t s o f
g
an
o rde r
y
y
m
h
m
ufi
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u
m
.
eq
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a
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l
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H e re t h e c ap it al le t t e rs s t a n d fo r t h e
de n o te d b y th e c o rre s p o n din g s all le t te rs , a n d a re t e s el e s
lb
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n
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i
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)
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s t h e e x a n de d
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as
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m
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h
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f m
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h u gh w my lw ys d v l p d t mi b y m s
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im
l
t
m
t
h
t lw y t h
p
i
ll
bj t i
t
y wh
mu h t fi d th lu
p
th
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t
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ig s
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A lt
e
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escr
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en
502
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e
a
a
a
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our o
e ter
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t
e,
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,
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a
s n o
a
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as
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o
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s
t
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es
e s
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e
n
n an
n
o
va
e
ts
o
n
s
e
se
f
o d,
e of
e ral
o
.
Th e
ex p an
de d
f m f th
o r
o
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6
0
1 2 3
s h
a
b
c
l s 2
e
l
bl
c
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2
62
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a
s
bs
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e
a
b
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s s 1
a
h lmti
m
de t e r in an t
l
2
c
b
s s s
“
6
0
3
1 2
u
f m h
c
b
s s l
h
f
s
f t re e a c to r ,
o n e t ak e n
ro
eac
ro w, an d o n e
ro
ea c
co l
n ; a ls o t h e
si n s o f
l
a
th e t e r s a re
a n d o f t h e o th e r h a l
Th e s ign s
g
o f t h e s e e ral e le
en t s
a
Th e rs t
y b e Ob t a in e d a s o llo
e le
e n t “ 6 0 1 in w ic
h
t
e s f x es
ll
o
o w t h e a rit
rde r,
e tic a l O
1 2 3
is p o s it i e ; w e S all c all t is t h e le a din
t
r
e le
e n t ; e er
o
e
g
e le
en t
a
it b y s it ab l in t e rc an gin g t h e
y b e o b t a in e d ro
ffi x e s
Th e Sign
or
is t o b e p re x e d to a n y e le e n t a c
an
a
ar
e
pp
t
at e a c
e e
en
s
t h e p ro d
a
f m h
m
h f
m m
v
m
h h
ufi f
v
h
h
m m
f m
u
su
fi
.
.
ct o
um
f
f ws
fi
hm
v y h
m
y
h
.
m
D
d du
E TE R M IN A N TS
4 25
.
f m
m
ro
h
en t b
an
t
e le a din g e le
e
ced
i
t
c an b e
y
g
t a tio n s o f t wo s ffi x es fo r in s t a n c e ,
b er o f pe r
e e n o r o dd n
ffi x e s 1 a n d
th e e le e n t a s b s c l is o b t a in e d b y in t e rc h a n gin g t h e
3 , t e re o re it s s ign is n e gat i e ; th e e le e n t “3 6 1 02 is o b t a in e d
f xes
rs t in t e rc h an gin g t h e s u fiix e s 1 an d 3 , a n d t e n t h e
by
1 a n d 2, e n c e it s Sign is p o s it i e
c o rdin
as
v
m
h f
fi
503
my t h u
m
s
u
w
i
f
a
h
su fi
.
h o s e le a din g
Th e de t e r in an t
b e e x pre s s e d b y t h e n o t at io n
Ei
su
m
v
v
h
.
a
mu
um
c d
b
1 2 3
4
e le
m
en
t is
b
c d
l s s
4
a
'
m
ju m
n t in dic a t in
h
l
a c e d b e o re t h e le a din
e le
e
t
e a
re a t e
g
g
gg g
p
ic h c an b e o b t a in e d ro
it b y
it ab le
o f a ll t h e e le
en t s
in te rc h an ge s o f s u fii x e s a n d a d s t e n t o f s ign s
2
th e
m
wh
mm
m
m
f m
su
.
m
my
l
o re s i
So e ti e s t h e de t e r in an t is s t ill
p
e le
e n t wit in b ra c k e t s ; t
t
h
e le a din
e n c lo s in
g
g
b a b c d
Is
se d a s a n a b b re vI a t i o n o f 2 =
s
s s s
u
h
e x pre s s e d
h us (
a
by
b
c d
1 2 3
4
.
d t mi t ( b d ) wh t ig i t b p fi d t
m
m
F mth l di g l m t b y p m
uti g th uffi f d d w g t
b
d
th i b y p m
uti g th uffi f b d w h v b d ;
; f m
by p m
uti g th uffi f d d w h v b d ; fi lly b y p muti g
th
u ffi f d d w b t i th qui d l m t b d ; d i
w h v md f u p m
ut ti th ig f th l m t i p itiv
E x a p le
I n th e
th e ele en t a4 b 3c 1 d5e2 ?
ro
a
c
4
er
e
x es o
a e
xe s o
e s
a
e o
r
o
e
er
a
c
2 3
1
er
e
4 5
S
a
e s
n
x es o
e s
n
e
a n
n o
e S
ons
n
s
o
x es
o
a
an
a e a4 3c l
re
e re
e e
c an
c
an
a
en
er
s
n
e s
e e
n
ro
e
l 5
2 3
ea
e
n an
e er
.
c
re
en
e
en
s
o
e
a e a4 3 c2
e
1
e
er
c
3 1
a4
xe
an
na
e
2 5
e e e
e
e
5 2
os
5
n
an
S n ce
e
.
u
504
I f in A rt 501 , e ac h o f th e c o n s t it e n t s b l , c l ,
k1 is
in o t e r wo rds
t o a sA l
e q al t o z e ro t h e de t e r in a n t re d c e
lb
it is e q a l t o t h e p ro d c t o f a 1 an d a de te r in an t o f t h e ( n
l)
in e r t h e o llo win g ge n e ra l t h e o re
o rde r, a n d we e as il
.
u
.
u
If
u
y f
ea c h
mi
deter
to
n an
mth
t is
th e
z ero
m
u s
m
f
c o n s ti tu en
t
h
t
e
p
ex c e
ts
fi
h
m
l m
.
f
o
th e
rs t, a n
d
fi
rs t
ro
if th is
mm
m
w
or
co u
c o n s titu en
m
f
n
t
is
o
a
equ a
l
deter i n a n t is equ a l to
ti es th a t deter i n a n t of lo wer
whi ch is o b ta in e d b y o i ttin g th e
rs t c o lu
n
an d
rs t
,
o rder
ro
f
o
m
e
m
fi
fi
w
.
s
u
u ms
f w h
u
ws
n
an
A ls o in c e b y s it ab le in t e rc h an ge o f ro
an d c o l
y
l
c o n s t it e n t c a n b e b ro
i
n to th e
rs t
a c e , it
o llo
s t a t if
t
p
g
an
ro w o r c o l
al t o
n h a s a ll it s c o n s t it e n t s e x c e t o n e e
y
p
q
z e ro , t h e de t e r
in an t c an i
e dia t e l
b e e x p re s s e d a s a de t e r
in an t o f lo we r o rde r
u
uh
um
m
m
mm
fi
u
y
.
h
m ms u fu l i
T is is s o e ti
o f de t e r in a n t s
m
.
e
se
n
th e
u
re d
c t io n
an
d
mplifi
Si
c a t io n
426
E
GH E R
HI
xa
m
pl
Fin
e.
d
v u
al
th e
Dm h h
u
h m
e o
AL
GEBR A
.
f
30
11
20
38
6
3
0
9
11
2
36
3
19
6
17
22
u
um
w
dm
u hm u
d u
n b y t i ce th e c o rre s p o n in g
i i n is e ac c o n s tit en t o f th e fi rs t c o l
n
co l
c o n s tit en t o f th e fo rt
co l
n , an d e ac
c o n s ti t e n t in th e s e c o n
n , an d
co l
b y t re e ti e s th e c o rre sp o n in g c o n s tit e n t in th e s eco n
we o b tain
1 1 20
5
8
an
d s in
th e
ce
0
3
0
0
15
—2
36
9
7
16
17
4
h
w h as t
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re e z e ro
8
20
5
u
c o n s tit
3
en
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h d
ts t is
0
1
0
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9
8
19
5
8
19
5
7
17
4
7
17
4
7
17
4
.
xa
ro
h
u
15
f ll w i
u fu l
P v h
505 Th e
c a s io n a ll
se
y
E
d
sec o n
20 5
8
3
d um
d
l
m
p
e
1
b
c
d
b
a
d
c
c
d
a
b
d
c
b
a
n
o
ex a
g
mpl
es
s
h
ew
a rt
—3
ifi c e s
t
n an
8
5
7
4
wh i h
Oc
a re
c
.
ro
.
a
dd
o
mi
ete r
e
t
at
—b +c —d
)
~
h
dd
w
(
—c — d
)
—b
a
h
d
.
in g t o get e r all th e ro s we s e e t at a + b
c
d is a fac to r o f th e
e ter
in an t ; b y a
in g t o ge t e r th e fi rs t an d t ir ro s an d s b trac tin g
ro
s we s e e t a t
fro
th e re s lt th e s u
o f th e
s ec o n
an d fo rt
b
c + d an d
b +c
a
d i s als o a fac to r ; s i ilarly it c an b e s ewn t at a
a co
a + b
d are fac to rs ; th e re ain in g fac to r is n u eric al, an d, fro
c
“
n ity ;
S i e , is e as ily s e en t o b e
p aris o n o f th e ter s in o l in g a o n eac
re s lt
en c e we
a e th e re q ire
d
By
m
m
h
a
u
m
m
Pro
.
v v
u d u
v h
e
t
1
1
1
a
b
c
d
a
a
3
b
2
b
3
c
c
2
3
u
m
h d
h
m m
u
.
(
a —
b)
(
a
—c
)(
a
—d
)
—c
—d
b
b
)(
(
)
(
c
—d
)
.
2
cl
3
d
v d m v h wh
um d
h
m
h
d m
d m m
um
mu
h d
u
u
Th e gi
h d w
u h w
h h
at
=
1
2
m
m
m
hv
E x a p le 2
h
h
d
d
in an t an is e s
en b = a , fo r t en th e fi rs t an d seco n
co l
n s are i en tical ;
en ce a
b is a fac to r o f th e eter in an t [A rt
Si ilarly e ac o f th e e xp re s s io n s a c , a d, b
c, b
d, c
d i s a fac to r o f
th e
in an t b ein g o f s ix i e n s io n s , th e re ain in g
e ter
e ter in an t ; t h e
fac to r
a co
st b e n
eric al ; an d, fro
p ariso n o f th e te r s in o l in g
b c 2 d3 o n e ac Si e , it is e asily s e e n to b e n ity ;
en c e we o b tain th e re q ire
re s lt
.
en
e te r
m
dm
h
m
.
m
m
vv
u d
4 28
HI
v
So l
12
th e
e
eq
03
.
dx
GH E R
ons
a
y+
+
-
l by +
v h
Pro
e
t
d
b
—
d
a —
a + b
16
Pro
.
— c —d
v th
a
b
e
at
2
2
(b
2
(
(a
a
b
2
2
0
wh
S
.
c
hw h
e
t
y+
+
2
+ by
3
a x
+b y+
u =
1,
.
cz = lc,
+
+ du
cz
=
3
3 =
c 2
lc3
lc,
2
c
( + d)
ca ( b + d)
ca — bd
a b ( c + d)
a b — cd
— 2 b — c c — a a — b a — cl b
)(
)(
)(
)(
(
bc —
a
d)
(
c
be
)
2
a)
2
b)
c
bc
ad
2
—
17
+ by +
at
—
+
ax
cz = lc,
b +c—a
c
13
1,
z =
ax
.
.
u ti
x
15
G
A L E BR A
ca
ab
b
(
— 0
)(
c
a
-
)(
Cl
—
b)
at
a
b
f
a
e
f
d
e
c
d
b
c
e re
2 —
=
C 0
f
d m
d
d h d
I f a e t e r in an t is
o f I ts
rs t, s e c o n
t h I rd,
,
t h e fi rs t , s ec o n , t i r ,
18
.
.
fi
o
2
+ 2 ae
f th e
ro
d
w
s
o r e rs ,
th
2bd
.
o rde r,
u
d if th e c o n s tit en ts
b e rs o f
are t h e fi rs t n fi gu rate n
S ew t at it s
n it
al e is
y
n
h
h
an
v u
u
um
.
.
XX XIV
C H A PTE R
M I SC EL L A N E O
US
.
TH E OR E MS A N D
E XA M PL E S
.
s h pt with s m m ks th
f mb i fly i w i g th fu d m t l
m
v
h d i t h u s f th w k
i
i
l
i
l
i
l
w
t
f
b
i
s
I th
d
507
g
p
p
p
p
u t t w d t l y d w w ms d
t th
ly t i lly
b gi
f m k wl dg f b t t
b t w
w id
wh i h
w p v
t i l ws f p
ti
b
l
A i h mti
p
y p ti u l
d th g
l th
y f
i
f v ifi t i
l
i
t
h
i
t
t
u
t
s
f
A
b
i
th
g
p
k
f
A
it i u u l t
i
h
m
i
l
A
l
d
H
Sy m
b
p
g
d t mk
di t i t i
b tw
t h m I th
b li l A lg b
m
b
l
i
i
i
f mw dfi
s
i
t
h
m
t
i
ll
y
i
ll
b
l
t
y
g
d du
fu d m t l l w f p ti ; i th l t
d th
w
s um th l w f A ith mti l A lg b t b t u i ll
wh t v th tu f th ymb l my b d fi d
u t b tt h d t t h ymb l i d h t
t wh t m i g m
l w
w t
Th u s g du lly
t h y my b y t h
d th
di
it s f
y A i h mti w u lt p i g p w l
lim
b
m
l
i
i
u
h
y
d
d
t
t
t
s
i
v
y
t
t
m
l
b
p
p
g
g g
l
wh i h w
t
t
i
h
i
i
d
i
i
t
l
fi
t m
d
t
h
A
t
t
p
g
s m tim f mth w y i wh i h t h g l l w f A lg b
u d f th i p m
t b li h d w
i
d
li
li y
d
i
i
v wh h y
u
t
t
t
t
i
t h mt i
pp
q
lly i t lligib l
i
508
C fi i g
tt
ti
i
v
i
v
u
t
l
l
f
p
g
ily
ymb l th f ll w i g l w
b li h d f m p i i
th
i h m i l d fi it i
WE
b egin t h i
e r an e n c e o f a lge b ra ic al o r
p
la ws wh ic h h a e b ee n e s ta b lis
506
.
t
r
on
e
er
a s
e
c ase s ,
e
e
a e
er
e an
o
a
e
as
e
a
e
a
a re
ca
t
,
t
en
e
on
e
o
,
a re
ass
e or
o
a re a
e
e
o
as
s
e
a
o
s
n
a
so
er
n
t
a
es n o
e
an
s
s
o
e
.
o
a n en c e
an
e
n e
,
on s
n
er
at e r
o r
en
e
e
ra n s c e n
u
r n
e,
an
n
e
e
r
e,
e n e ra
o
e
a
n
n
on
e r
s ra c
.
n
o s
n a
e
c
re
s
an
eo r
e
a
,
e r re a
n
n
a
e
.
o
e s
re s
n
an
a
e
o
ro c e e
a re c a
on
ra
s
o
e
n e
c,
a e
t
s
e
a
ca
e ra
e
.
o
een
e
o
or
en era
ra
a
e ra a n
e
ca
e
ac
on
ra
our a
n
e
e
e
e
en
a
n a
c
et c a
s o
a
e
e
ra
e
an
ar
un
e
.
n
o s,
et ca
e
,
a
e
e
e
en
e
.
e s
e
en
e
on
e n se a r
a
n e
e
t
nc
.
t
o
co n
e
n
r
ro
s
s
r
s
n
e a
a
e e
o
es a
v e rs a
ar
es e
o
s
re
e ra
e
n
n o
c en c e o
r
s
-
n ar
e,
o
e na
n
e re n o
c
s
o
o
e a
a
o
our
e
o
a
n
e
r n c
on
ar
e
r e o
a r c a s e , an
c
ea
a
n
a
e
o r
o
S
o
ce
e
a
ou
o
n o
a
es
n e o ur s
e
en c e
an
ar
a
e ra , a n
ca
or
s
s
en c e
o
n
ra c a
ro
n
e rat o n s c o n s
es e O
o
eve r
n
e co
e
a
e cer a n
ro
e
ca
er
o
c
e
re v e
n
e
e re
o
r e
,
e
e
er
a
c
o
se
e
u
eas ,
ne
e o
: a
ca
on
os
ex
e
n
.
a
s h a ll
o
n
o
n
on s
.
en
a
on
to
s a re e a s
os
t
e sta
e
n
s
e
e
ra
ro
a
a
es
o
r or
4 30
HI
I
mmt
Th e L aw o f Co
.
A ddi tio n s
1
()
hu
T
s
an
u
hu
T
at io n
,
b
—
o
a
my
b
c +
b
:
bs
a x
— c + a
my
II
—
z
b
-
d
su
b
-
r
b tra c ti o n s
hu
T
(
s
o rder
an
y
in
an
.
a
e
o rder
y
.
c
my
d
so
on
.
—
z
b
(
=
wh i h w
e en
u
f ll w
i
n c ate as
b e di s trib u ted
a
—
a
—b
a
)(
d)
c —
o
o
s
o ver a dditio n s
v
u
o f I n dic e
i
()
a
mX
m m
+ c
bc
mt
A n d s in c e divis io n is t h e re
b u tiv e la w fo r divis io n re q ire s
Th e L aws
b
= ac — ad —
S
ee E le
[
.
s
.
(
III
b
x
d divis io n s
an
o
;
a
Th e Law o f D is t rib u t io n ,
.
M u ltip li ca tio n s
an
x
a
c
o
.
md
be
a
in
e
a
f ll w
n c ia t e a s
md
an
a
u
e en
be
a
d divis i o n s
an
s
—
.
c
b tra c ti o n s
z
G E BR A
wh i h w
su
M u ltip lic a ti o n s
ii
( )
AL
d
+
a
GH E R
s
n o
se
p
bd
.
mu ltipli
f
e rs e o
+
A lgeb ra , A rt
a ry
en
,
c a t io n
dis c
a ra t e
u
s s io n
s
3 3,
.
t h e dis t ri
,
.
.
a
n
“
m+
mt
S
e e E le
[
en
1;
A
l
y
geb ra , A rt 23 3 t o 2 3 5 ]
ar
.
h l w l id d w fu d m t l t
u b j t h vi g
i
h
i
b
t
h
v
u
h
y
s
y
d
t
t
m
b
l
m
l
d
p
p
p
p
l
i
i
d
t
i
d
t
v
h
hy
i t d i
uh wy
p
g
h t th p ti b v i di t d
i h mti lly i t lli ib l
g
I f th
di i
d
t h ld b y t h p i ip l
f Sy m
b li l
w
Alg b
u m t h l ws f A i h mti l A lg b
b t u
i
d
h
t
t
i
v y
wh
h
t
i
t
i
th i
mp
p
p
i
By hi
l d
u w
u d h t th l w f
A lg b i l p
i
lf
i t t
d t h t t h y i lu d i
h i g li y h p ti u l
y A ith mti
f di
509
F m h l w f mmu t ti w d du th u l
f
h
mv l d i t i f b k t [E l m y A lg b
A
2 1 2 2]
d by h
id f h
ul w
b li h h l w
T
ro
e en
os
t
es e
t
t
e
ra
e r
n
t
rt s
.
e a ss
e
e n e ra
t
t
re
,
o
e
t
ro
a
o
an
an
e
a
c
t
ter
-
t
re t a
e a re
c o n s s en
e a
co
on
ra c
t
e
e
n
es
ass
re
an
es
e
e
to
ra
.
ca
e
r
e
s a ss u
e
e
a
ce
c
e
ta r
e e s ta
s
n c
e
e
en
a
o
r
e
a
o
a
e
e s
n
n
e
c
a
n ar
on
s
c
t
a
,
o
ca
e
o
es e r
e
ca
or
a
o
o
t
on
,
ec
o
r n c
e
a r c a se s o
o
o ours
s
e
e
r
n
a
a re a r
,
rs e
co
en
a re re s t r c
e
o
a
n s er
t
t
o
a re s e
ar
a
ca e
e
s
a
on
n
n o
e
n
a
e
o
e ra t o n
o
an
,
t
t
os
ac c e
.
as
n
s
on s a
on s
u s
o
ra
t
ra c a
e
te
e ra
s
.
o r
e
c a s e an
ea
on
on
co n
er
e
t
e
e o
ese
n
a
e an
a
e
s a re
a
s
O
e
n
.
r
es
e ra ,
t
e
a
4 32
HI
s in c e
A ls o
h
t
co
o se
mpl
a
bo
v y
v m ti
u mb i i
ex n
o n e d,
en
e
s
er
n c
it
g
on
.
a
a co
e
o
n
s
o
o
e n era
n
G
A L E BR A
fu ti i v lv s
i
f ll w t h t
l
mpl u mb
ra t io n al
er
e
GH ER
n o
o
ra t o n a l
a
ra t io n
e
p
fu
i
n ct o n
s
but
o f a
er
ex n
.
f m
fu y
m y
hu
h m
y h h u h fu
u
um
f m
m
i
i
f
i
l
i
t
h
m
u
u
d
d
l
Th
g
p
f m b t t h md f t i g it i w hy f tt i
i A t 220 h
W h v
wh i i fi i
m
l
u
i
y
u
y
y
b i g
t
h
i
m
b
i
m
i
l
l
t
;
y
q
q
y
d fi d by m
f th
u
i
t
q
m Lim 1
wh
i i fi i
dy b i g
u
i
i
l
t
y
q
Th d v l p m t f h t h
y f mpl u mb w ill b
f u d fu lly di u d i C h pt
d
f S h lOm
il h
w
”
lo g ( x
a
,
+ iy
c an n o t
b
ll
e
E x p res s io n s o f t h e o r
)
’
t re a t e d wit o t Trigo n o e t r ; b u t b y t h e a id o f D e M o ivre s
n c t io n s c a n b e re d c e d t o
t e o re , it is e a s t o S ew t at s c
A
l
b
e rs o f t h e o r
iB
ex n
co
p
.
re ss o n
e ex
or
a
”
‘
e
“?
s o
er
an o
u
e s e en
a
e
x
r
nc
e
re at n
e o
o
n
rs e
co
t
.
e
n
o rt
s
o re
o
e n e ra
en
a
t
on
.
at
n
en
x
e
re a
an
e n
e an s o
n e
t
an
t
e e
e
s
n
n
t
an
n
te,
?
e
e
a
ar
s
on
a
71
.
e
x an
e n
e
o
n
H
an
e
"
en
re a
an
e o
en
sc
db u ch der
t
o
sse
e
es
An
b
O
a ly s is
co
X
e rs
a
.
mh
51 4
f
o
x
is
Le t
f (x ) b y
x
; le t
mi
To fi n d th e re
divided b y X
a
f( )
x
x
Q
a
den o t e
n t il a
u
b e th e
a n
an
w
Q
is
fi
n
’
‘
o
.
c
c
s
.
an
y
s an
n
d
u
se
ex a
mpl s illus
fu l i
e
n
ro
p
vi
n
g
.
ra ti o n a l in te ra l u n c ti o n
g
f
rat io n a l
y
m
u
v v
m
fu
wh h
w ill mi u
z th
it e fo
r
fi
n
it e
it
x
re
a,
f (a)
no
e
v
in t e gral
n c t io n
o f x ; di ide
re
a in de r is o b t a in e d
ic do es n o t in o l e
o
t
i
n t a n d B t h e re
e
ain de r ; t e n
q
,
v
u
der wh en
e rs
.
Sin c e B do e s n o t in o l e
al e w e
i
g e to x ; pu t x
v
te,
n
XI
v m h m
wh h w f
fud
h y
u
.
n
ex
an
51 3
W e s all n o w gi e s o e t e o re
t ra t in g
e t o ds
ic
ill o t e n b e
o
ide n tit ie s , an d in t h e T e o r o f E q a tio n s
.
n
.
eo r
n
a lge ra i s ch en
h
t
an
s
n
v lu
a
a n
en
Q
x
es o
0
f x,
B
h
en ce
v v
h
n a lt e re
d
wh t v
a e
er
M I S C E LL A N E O
US
E X A M PLE S
TH E O R E MS A N D
y v
4 33
.
h
h
a, t en R
0, t a t is
I f ( x ) is e x a c t l di is ib le b y x
C OR
e n c e if a ra ti o n a l i n tegra lfu n c ti o n of x va n is h es wh en
a
0
( )
a
a , i t i s di vis i b l b y x
x
f
f
h
.
z
e
.
Th e p ro p o s it io n c o n t a in e d in t h e p re c e din g a rt ic le is s o
51 5
h ic h a s t h e a d a n t age
l t a t w e gi e an o t e r p ro o o f it
se
o r
o f t h e q o t ie n t
o f e x ib it in g t h e
u fu h
v
h
h
f m
S u pp s t h t t h fu
f
.
o
v
.
is
n c t io n
e
a
e
u
w h
o
f
m
di
n
e n s io n s ,
an
d le t it b e
de n o t e d b y
po
t
h
en
th e q
u
x
”
+
ps
w ill b
tie n t
o
pl
x
gm
mi d
n
—l
gl
e
le t B b e t h e
re
+
(
u
y
ps
d eq
an
w e h a ve
—
92
qs
—
B
u
+
t a in in g
co n
p
q
h
en
n
" 2
qlx
p
a
q
R
“
i g th e
s
t
x
p
de n o t e it b y
e n s io n s
+
p
or
9.
or
p
ts
92
3
2
91
+ 17
o r
Qs
z
a
qe
+ 23
or
B
z
a
g?
s,
qsx
fi c ie n
c o ef
at n
z
+
—2
)
11
ps
x
m
+
04
— a
q
t
" 3
“
1 di
x
a
x
n
2
"
fn
x
er n o
a n
M lt ipl in g o u t
e o
x
”
n ,
q
k
f li
e
o
p
w
e rs o
f x,
+p
—l
n
o
,
i
ffi i t i
th
t
i
h u s h u s iv
u
t
f
m
d by
q
ffi i t l t f md
mu l iply i g b y th
d ddi g t h
f fi di g t h
ffi i i th div id d Th p
t
mi d my b
u s iv t m f t h q u t i t d t h
hu
d
t
g
t
s
eac
t
n ex
s
cce s
c en
co e
cce s
e
n
o
0
21
a
e
.
e
,
ro c e s s
re
e
an
en
ge
P2
Th u s
R
ag — + )
lt
1
l
:
-
p0
a
a
.
Q2
(6
s
an
o
a n
mu
H
.
A
.
+
—
a
2
+ p sa
v
.
qQ
P
(6
sa
9
11
—2
q —l
Q3
(M
I f t h e di i s o r is x + a t h e
t is c a s e t h e
ltiplie r is
a
h
P3
a
91
.
o r
en
s
To
H
o
as
en
e
e
n
c en
e
o
s
er
e
a rran
t
c en
co e
e
a
n
co e
e
n
p —l )
"_ 2
+
m mt h
e
p
n
e
o
p
n
.
n
d
c an
be
o r
a
n
er
e
n
e
n
e
a
e
4 34
E
xa
m
pl
dvd d
i i
is
H
e
by
th e
ere
Fin
e.
x
+2
d
th e q
u
multipli
er
hu
th e q
s
der is 1 1
a n
u
o
d re
tien t
an
2,
d we
is
1
6
G E BR A
mi
a n
.
d wh
er
en
an
3 x7
x6
5
hv
a e
31
0
14 — 28
0
6
12
21
— 24
3
6
12
3
—7
3
T
o
AL
.
3
mi
GH E R
HI
14
tien t is 3 x
6
7x
5
l 4x
0
4
3x
3
5
6
11
2
6 x + l 2x
3,
an
d th e
re
.
m
w
o rk h a s b e e n a b ri d e d
I n th e p re c e din g e x a p le t h e
51 6
g
t h e c o e fii c ie n t s o f t h e s e e ra l t e r
z e ro
n on l
rit in g do
by
,
s c o rre s o n din
s e d t o re p res e n t t e r
c o e f c ie n t s b e in g
p
g t o p o e rs
T i
e t o d o f D eta ch ed Co
c
i
e
n ts
a
ic a re a b s e n t
o fx
efi
y
s a e la b o r in
e le
e n t ar
a l e b ra ic a l
sed t o
e
b
en tl
re
g
q
n c t io n s w e a re de a lin
en t h e
w
i
t
ro c e s se s , p a rt ic la rl
g
p
Th e o llo win g is a n o t e r ill s t rat io n
a re rat io n a l a n d in t e gral
w
w y
m
fi
u
wh h
hsmh
f u y u
v
u
u y wh
fu
f
m Dvd
.
m
.
E x a p le
i i
.
e
3x
8x
4
-
5x
?
6
x
2
3
h
26 b y
33 x
w
m
y
.
5
ms
v
x
3
h
u
2x 2
.
4x
8
.
8
—2 4
7 + 2 —33
4
—2 +
—2
6
3
3+
hu
u
h ud
T
th e q
s
o
tie n t i s 3 x
2
d h
2x + 3
an
— 24
d th e
d
h d h
m
h u
ill fu t h
Th w k my b
51 7
i
i
k
m
wh
h
w
H
g
mi
re
a n
w
d
dv
er
is
5x
2
.
v
g o wn th e i is o r, th e Sign o f e ery
ter ex c ep t t h e fi rs t h as b een c an ge ; t is en ab le s u s to rep la c e t h e p ro ces s
s c ces s i e s tage o f th e
of s u b tra c ti on b y t h a t of a ddi ti o n at e ac
or
It
l
o
s
e
D ivis i o n
e n o
e
.
a rran
b
en
t,
tic e
or
t
at
e st
a
s
c
ritin
in
n o
r
as
n
v
w k
idg d b y t h f ll wi
.
e r ab r
’
o rn e r s
e
e
M eth o d of
o
o
g
Sy n th etic
n
.
1
2
4
3—8
6
3
—
2
um
12
4
+
3
24
8
6
0
+
16
12
5+
u
24
2
v
Th e c o l
n o f fi g re s to th e le ft o f th e
ertic al lin e
c o n s i s ts o f th e c o e ffic ien t s o f th e
i iso r, th e Sign o f e ac after th e fi rs t b ein
g
c an ge ;
th e s ec o n
o riz o n tal lin e is o b t ain e
by
ltip ly in g 2 , 4 ,
8
b y 3 , th e firs t t r
o f th e q o tien t
e t en a dd th e ter s in th e s ec o n d
W
e
co l
n to th e ri g t o f th e
ertical lin e ; t is
2,
ic i s th e c o e ffi
gi es
c ien t o f th e s ec o n
o f th e
ter
o tie n t
Wit th e c o effi cien t t s o b tain ed
[ E xp lan a tio n
h
d
um
dv
d
h
m
u
h m v
d
u
q
.
.
d
h
h v
h
h
mu
m
wh h
-
hu
HI
436
Pu t
p
z
a n s io n
0 th
:
,
y)
f (x
o
6
.
fi c ie n
c o ef
t
f
o
x
z
n th e
i
y
ex
3
Pu t
B
G
A L EBR A
3 , b ein g t h e
A
en
GH E R
6)
d we ge t
an
w
B ;
+
h
en c e
.
hu
T
(
s
x
x
z
y
3
y
3
fu
)
3
2
?’
z
3xy
3x y
A
52 0
a ria b le s , w e n it s Sign b u t
e o f a n y p a ir o f t e
c an
g
v
h
h
.
hm
a
a re alt ern a t in
t it s
n o
fu
g
i
h
Sz
3y x
us
3z x
x
2
6xy z
.
h
wit re s p ec t to it s
a l e is a lt e re d b y t h e I n t e r
an d
a
s
y
v u
—a
b
(
s
n ct o n s
v
Th
.
g
a ltern a tin g
to b e
s a id
is
i
n ct o n
2
3s
2
)
+ e
g
(
b)
a
.
h
fu
u
mu s
n c t io n
I t is e ide n t t a t t e re c a n b e n o lin e a r a lt e rn a t in g
o re t a n t wo v a ria b le , an d a ls o t a t t h e p ro d c t o f
in o l in g
n c t io n
t b e an
n c t io n a n d a n a lt e rn at in g
a s
e t rical
a lt ern at in
n c t io n
g
v v m h
s
h
ymm fu
fu
fu
mt i l d l
ti
s my b
52 1
Sym
ti g fu
i ly d
t d by w i i g d w
d p fi i g
f th t m
th
ymb l 2 ; t h us E t d f th m f ll th t m fw h i h
i t h t yp 2 b s t d f
th
m f ll th t m f wh i h
b i th
yp ; d
if t h fu ti
i v lv s
F i t
f u l tt s b d
.
e r ca
.
c se
en o
s
o
an
r
d
e,
e
an
er
a,
c
so o n
n
on
so
m
y if th fu
e
2 a (b
c
)
= a
Ea b c
on
h
s
or
o
e su
o r
s
er
e
e su
or
n c t io n
g
so
o
e
a
o
e
a
re
an
x n
s o
er
er
e
n s an c e ,
co n
e
a
c
s o
n c
on
c
n
o
e
,
2
d
n
on
n c
n
on e
o
.
a
.
Si ila rl
an
s
an
t
e
e
a
t e rn
a
a s an
e
s
a
t
r
o
e s
a
e
an
a
in
b
(
g
v lv t h
o
)
c
g
bc
b
es
b
g
e
(
e
—
g
ca
c a
s
le t t e r
ree
a
)
c
g
(
a,
b,
c,
b) ;
a
b;
.
h
h
u
I t s o ld b e n o tic e d t
f
Ea b do es n o t c o n s is t o f t
g
Ea b
m
my
m
Th e s y b o l 2
re a rd t o t wo o r
g
Ey z ( b
a
2
a
b
at
ree
2
a o
a ls o
o re s e t s
h
w
en
h
e re a re
mb
ter
b
t
z
s,
c
u
b
u
t
g
o
o a
:
2
0
t
6
s v lv d
le tt e r in
re e
hus
o
e
.
mply ummti
b e se d to i
o f le t t ers
; th s
u
h
f s ix
g
a
t
s
a
—b
)
.
on
wit
h
N
M I SC E L L A
f m
or
v
s
52 2 Th e a b o
t h e p ro d c t
.
u
(
(
(
e n o
m
E x a p le 1
(
D
a
t
e
+ b)
to
us
y mmt i
E X A M PL E S
ex
Ea
a
3 Ea b
6 2 ab c ;
“
42 a b
2
)
Ea
”
2a
4 37
.
an a
b ridge d
e r c al e x re s s io n s :
p
Ea
4
in
re ss
p
s
d)
v h
Pro
.
fs
o
er
3
Ea
x
w s
)
c
le s
e n ab
3
b
Ba
TH E OR E M S A N D
tat io n
c
c
a
US
d po
an
b
a
b
a
EO
32a b
2
6abc ;
2
3
6 2 6t
hu
t
s
2
1 22 a bc ;
’
3
Ea b
.
at
5
a
5
b
5=
(
5a b
b)
a
(a
2
+
h
ab
+
u
wh h
th e le ft b y E ; t en E is a f n c tio n o f a
ic
ilarly b is a fa c to r o f E
e n ce a is a fac to r o f E
si
en a = c ,
an is es
b , t at i s a + b is a fac to r o f E ; an d t erefo re
en a
A gain E an is e s
Th e re ain i n g fac to r
s t b e o f two
E c o n tain s a b ( a + b ) as a fac to r
it res p ec t t o a an d b , it
st b e
etrical
i e n s io n s , an d, s in ce it is sy
2
2
s
o f th e fo r
A a + B ab + A b ; t
en o
ex p re ss io n o n
te th e
v h wh
h
v h wh
dm
h
(a + b )
wh
u
wh
A
ere
an
dB
mm
u
en ce
a
= 2, b
m
E x a p le 2
Fin
.
3
(b + c
D
3
en
e
t
a e
e
t
an
d
5=
d
w h v
w h v
d hu
1,
A = 5, B = 5;
b
ep en
P ttin g a = 1 , b : 1 ,
p ttin g
a
5
o
fa
)
c
)
(
+ h) (A a + B a b + A b
an
db
2
)
wh
,
c
3
o
req
2B ;
ui d u
re
res
lt
at o n c e
fo llo
+
d
3
)(
a
c
(
)
h
h
w
s
.
f
a
—b
u
th e expres s io n b y E ; t en E is a f n c tio n o f
b as a fac to r [A rt
en a = b , an d t ere fo re c o n t ain s a
a; t
s E c o n tain s ( b
c
c an d c
co n tain s t h e fac to rs b
)
facto r
en o te
mu
.
1 5: 5A
th e
th e fac to rs
(b
a
2
.
1 5= 2A + B ;
a e
s
(
ab
h
mu
wh
hu
5
d
in
are
m
.
m
m
hu
)
.
a
.
(c
wh h v h
ic
Si
an is
mil
a
)(
arly
b)
a
es
it
as a
.
u hd
A ls o Sin ce E is o f th e fo rt
fi rs t egre e ; an d s in c e it is a s y
M ( a + b + c)
th e fo r
[A rt
d
m
.
m
mu
th e re ain in g fac to r
s t b e o f th e
e tric al f n c tio n o f a , b , c , it
st b e o f
mm
e gree
u
mu
.
E
:
m
M (b
—c
)(
c
—a
)(
a
—b
v
)
v u h
mt
ay gi e t o a , b , c an y
To o b tain M we
al e s t at we fi n d
s b y p ttin g a = o , b = 1 , a = 2 , we fi n d M : 1 , an d we
ven ien t ; t
req ire res lt
hu
u d u
m
u
hv
co n
a e
th e
.
E x a p le 3
.
-
D
o s
hw h
S
5
x
e
t
at
( 2 + x)
h
v h wh
— z
en y :
te th e exp res s io n o n th e left b y E ; t en E
an is e s
,
ilarly z + x an d x + y are fac to rs ;
an d t ere fo re y + z is a fac to r o f E ; s i
Also Sin ce E is o f th e
t erefo re E c o n tain s (y + z ) ( x + x ) (x + y ) as a fac to r
h
en o
h
m
.
438
hd
fi ft
sy
GH E R
HI
mi
th e
mmt i
egree
e r c al in x ,
y
,
2,
G EBR A
fac to r is o f th e
s t b e o f th e fo r
a n in g
re
AL
mu
it
.
m
s eco n
dd
s in ce
egree ,
A
hu
Pu t
w
en c e
an
d we
hv
hu
= 0; t
z
3 5= 5A + 2B ;
s
s
ire
req
We
re s
t
.
f
s
s h h
my
h e re fo r re e ren c e a li t o f ide n titie w ic
an
l in t h e t ran s o r a t io n o f a lgeb ra ic al e x p re s s io n s ;
h ave o c c rre d in C ap XXI X o f t h e E le en ta ry A lgebra
.
se
f th e
1 O= A + B ;
u d ul
th e
a e
u fu
a re
s
A = B = 5,
523
o
=
1
,
y
x = 2,
pu t
h
t
e
c o lle c t
f m
h
u
—
b
b
c
c
E (
)
2
a
:
m
.
—c
-
c
-
a
)(
—b
.
)(
)
— c c —a a — b
b
(
)(
)
)(
—b
c (c — a
a
)
)(
)
— c c —a a — b
b
(
)(
)(
)
b
(
=
—c
b
(
)
2
.
a
.
.
-
.
b
(
a
3
b
+
3
+ c
s
Th i ide n tit
a
3
+
b
3
+ c
b c — ca — a b )
s
y my b
a
3
3a b c =
e
i
g ve n in
é
(
a +
b
+ c
an o
t
h f m
er
(
—c
b
(
(
a +
b
+ o
)
s
—
a
Eb c ( 6 +
3
—b3
0
)
or
b
{(
)
2ab c = ( b
2 a b c = (b
.
,
c —a
—a
)
9
+
(
—b
)(
)
)(
c + d
(
)
a
a + b
)(
)
b)
c
a
b)
a
2
.
.
.
— abc =
2b
2
c
2
2c
"2
” z
2a b
a
a
4
b
4
c
4
—
E XA M PLE S
1
by
x
.
+5
my b
a
e
F
in d th e
v
re
di is ib le b y
mi
a n der
x
3
.
h
w
en
.
XXXIV
.
c
)
.
a
.
3x 5+ l 1 x 4 + 9 0x ?
1 9x
v
53 is di ided
HI
4 40
GH E R
G E BR A
AL
.
25
.
(
2
26
.
x
s
u
Pro
27
b s tit t e 3
.
u
v h
t
e
3
3
+y + z
b,
a
e
fo r
c
s
d
th e
v lu
3
.
b
(a
2
(
b ) (a
a —
2
c
)
.
u
3 a b c is
3
f a +b +c
a , b , 0 re s p ec t i
v ly wh
e
,
ere
-
c
)(
x — a
2
—
b
0
(b
)
2— 2
b
c
2
(
b)
2
(
c
-
) (b
b
(
a
(
Ifx
b)
-
(
—c
)(
a—
—c
)(
x —
b)
2 —
2—
c
a
)
(b
23
d)
b
(
c
2 s
p ,
t
e
e
b)
c—
(
x — c
)
2
(b + x)
hwh
y
N
EO
US
52 4
M an ide n t it ies c a n b e
u s e o f th e
ro
rt ie s o f t h e c b e
e
p p
2
ill b e de n o t e d b y 1 , w, ( 5
u
w
(
—b
c— a
a — c
u
c—
q)
b) (c + x)
a —
d)
a+ n
)(
u
)(
0+
‘
at
D E N TI TI E S
I
y
re a dil
e s t ab
f
u
v h wh
en
ro o
ts
o
.
n it
lis
y;
h
ed
as
by
u
s
u
mki
a
al
t
h
n
g
e se
.
m
E x a p le
hw h
S
.
Th e
en c e it
u
t
e
(
h
)
)(
d
+
29
M I SC E L L A
.
— a
32' E
d xy z
z = s , an
y
P
a
if we
c
—a
— a2
b cd
a
d
.
(
a
n alt e re
b
a
a —
.
f
e o
a
3
o
33
Fin
3 a b c)
—
3
v lu
th e
at
s
a
3 —
b
3 a bc)
+c
+
3
— c S h ew th at
,
b+c
If x =
a
3
x
e x p re s s io n
mu t
s
co n
,
at
+ y)
E
tain
7
-
hv
taI n s
E
x
co n
2
wy
tain
es
x
= 0,
=
y 0,
fac to r
x
+ y = 0;
.
— w7 — 1
}y
fly
(
z
-
wy
(
x
as a
h
we)
1 ) y?
w— 1 1y7 = o ;
m
my
facto r ; an d si ilarly we
fac to r ; t at is , E is i is ib le b y
s x
as a
as a
an is
a e
w
n ce
=
7 xr
y
(x + y )
r-
he
7
th e left
on
,
xy
P ttin g x z wy , we
— x7
dv
(
5
6
?
o r at
m
y+y
a
2
.
s
hw h
e
t
at
it
co n
M I SC E LLA N E O
u h
F rt
E b ein g o f
ain in g fac to r
m
th e
re
(M
u
wh
y)
s
= 2,
x
e
7
—
=x
96
y
y
u
y
a e
We
52 5
.
s
a
e
a
a
9
b
+
s
+ y)
x
kn o w
2 1 = 5A
a e
2B ;
— x7
7
f m l m t y A lg b
ar
en
e e
ro
b
in E x
e s ee n
g
+ e
3 , A rt 1 1 0, t
.
.
3
3a b c
0
+
b
+ c —
a
3
m
3ab c
hw h
S
.
t
e
a
du
3
at
(
s lv
be
c an
re o
ed
— ab ;
)
cd
a + a
in t o t
h
2
b
4-
wc ) ;
lin e a r
re e
f
ac t o rs
;
at
-
th e p ro
3
+b +c
mA
h
(
a
+ab +x
h
b tain th e t
o
es e s ix
’
c
)(
3
3
c
3 ab c
o
an
C3
f
d
x
3
+y + z
3
3x y z
3A B O
.
) (a wb Me) ( a w b we )
2
z
x
w
w
w
x
+
+
+
)
(
(
y
y
c
fac to rs in th e p airs
x
3
2
wz ) ;
+ wy
a
p artial p ro
ree
du t
3
B
+
b
(a
ct
By takin g t
du
an
d
(
(
a
b
a
c
)(
+ wb
(
.
y
x
we)
2
z
);
2
w z ),
wy
x
c ts
2
A + w B + wC ,
A + B + 0,
ere
hu
T
th e p ro
s
du
ct
(A
:
—A3
+B
(A
C) (A + i
3
+B +
3
C
3213 0
v
2
(0
3
x
C)
.
u
s
s
v v
u
I n o rde r t o fi n d t h e al e o f ex p re s s io n
in o l in g
52 6
b , 0 w e n t e s e q an t itie s a re c o n n e c t e d b y t h e e q a t io n
ig t e plo t h e s b s t it t io n
0, w e
b c
.
a
— bc —
4
b e p u t in to th e fo r
Th e p ro
a,
h
+
0
2
—
b c cd
—
2
3
3
2
t h at
ra
e
0
b
E x a p le
wh
e n s io n s ,
s
a
c an
i
hu
3
3
hv
we
en c e
b
+
a
h
thu
dm
A = 7, B = 7 ;
(
al o
441
.
2 1 : 2A + B ;
en c e
d
E N TITI E S
y)
(M
hv
w h v
1,
D
m
an
,
7
7
I
d xy (x + y )
o f fi ve
2
2
o f th e fo r
A ( x + y ) + B xy ; t
s
se e n
P ttin g x = l , y = 1 , we
p ttin g
we
v
mu t b
er,
US
h
h
u
mh m y
a =
+ lc,
h
h w v th
m h d h ib it d i
If
et
o
o
ex
e
er
e
b
u
2
= wh + w 1c
,
th e
c =
s i v lv
f ll w i g mpl
e e x p re s s io n
n
u
o
o
n
n
o
exa
e
2
w h + wk
.
y mmt i lly
f bl
b, c s
e is
re
p
a,
e r ca
e ra
e
.
the
44 2
HI
m
E x a p le
If
.
hv d
We
a e
i
a
b
6
5
(
a
O, s
c
+b
AL
hw h
t
e
(
5
a
3
GE BR A
.
at
b3
+
+
c
)(
3
a
2
+b +
b x ) ( 1 + 023)
= 1 + p x + qx 2 + rx 3 ,
r= ab c
ere
H
2
tically
en
1
1
a
x
+
(
)(
wh
GH E R
u
en c e ,
g th e
sin
d
co n
v
itio n gi
en
,
2
l
x) ( 1 + bx )
a
+
(
hm
Taki n g lo garit
—1
n
l)
"
"
n
b
a
c )
(
s
d
an
eq
u
+ qx +
atin
(
2
)
qx
u
By p ttin g n = 2 , 3 , 5 we
a
2
2
+b +
c
(
3
x
r
o
qx
2
.
ts
e xp an s io n
n
" in
c o e fficien t o f x
3
rx
fi c ien
c o ef
th e
g
"
t o f x in th e
co e fficien
.
1
)
3 2
rx
(
$3
qx
o
of
a
3
3
+b +
wh
an
en c e
u d u
ire
req
lt
res
c
h
(6
3
3
+ b +c
fo llo
w
s
q ";
—d
mp
are
°
2
3
a
.
~
an
4
3
d
co n
s atis fi e
itio n is
d h
;
en ce
fl) }
-
l
5
+
(
a
-
fi)
E x 3 , A rt 522
.
.
5=
5 (IS
r) (r
-
If
(
a
+ b
)
c
—
I9 )
3
—
l {(fl
2
B) }
.
e
b
(
S
h w th
e
3
3
“ +
54
3
-
3
0 ,
XXXIV b
.
.
S
y)
c
t
e
3
b
3
(
h w th
e
— l) — c
hw h
hw h
l(
—8 —
7
17
06
a
—a
fi) ;
at
.
h
w
en
n
is
a
v
Po s iti e
at
n —
n
S e
t at (x + 3/l
x
o dd
o
s
i
t
i
e
i
n
t
e
p
ger n o t
S
d
.
t ege r
is
en
is ,
1
n
v
5
E XA M PLE S
1n
)
.
5
at
3
rx
3
v u
—
2
3
'
:
z
b
c
a
a
,
,
y
B 7
(3, th e gi
i en tically fo r all al e s o f a , B, 7
{ (B
t
a
at o n ce
=
a e
qx
a e
3
hv d
we
co
a
5
5
d th e
If
6
+b +c
lo g ( 1
hv
we
)
3
2
a
,
b tain
2
5
x
3 3
rx
2
”
5
f
”
v
—
a
"
c— a
is di vi s ib le b y
ltiple o f 3
mu
y
)
—
2
b
(
xy
)
—a — b
2
c
(
)
(
2
5
0
+W
y
.
2
if
.
at
cx
az
)
3
c
3
(
a
y
bx )3
3 a b c ( be
y)
c
( cx
az
) ( ay
bx )
.
44 4
HI
If a +
a
23
+
d
5
+c +
t
e
5
5
b
.
24
hwh
s
5
GH E R
a
+b +
(
.
a
b
+d
3
c3
a
2
d
)
25
.
+b +e
2(
26
27
S
.
S
.
28
hw h
t
e
t
v
R es o l
.
an
b)
s
(
at
hw h
e
+b +c +d
(
d 20
s
c
2
)(
3 xy
6x y
2
+b
a
a
2
3
x
a
2
2
2
)
y
)(
ad
9 (be
2
0 ,
p
ro
5a b cs
)
(y
3 3
)
2
abc
da b
=
a
2
2
2
2
9 ( b cd + cda
3 2
3
c
3
If 2s =
.
3
5
3
GE B R A
at
3
3
AL
(
3
v h
t
e
3
2
at
(
a —
b)
(
a
—
)(
e
a —
4
3
(
cd
b
)
.
2
3 3
x
)
2
2
(
(
a
at
2
6W
b d)
ca
3x y
x
(
+ 2)
r
2
as
«
+ 23
3
.
d)
in to fac to rs
e
(
2 2
2a b e
2
a
b
3
3
3
0
b
a bc
)
E L I M IN A TI O
h
3 3
c a
3 3
N
c
a
3b 3
.
.
hv
m
h
I n C apt e r X XXI II we a e s ee n t a t t h e e li in an t o f
a
a t o n c e b e writ t e n do wn in t h e
o f lin e a r e q at io n s
a s st e
y
in at io n a p
o f e li
e t o ds
G e n e ral
o f a de t e r in an t
or
o n d dis c s s e d in
o f an y de gre e wi ll b e
i
e q at io n
l
c ab le t o
p
a
in partic la r we
re e r
t re at ise o n t h e T e o r o f E q at io n s
y
’
t h e s t de n t t o C apt e r I V an d V I o f D r Sal o n s L es s o n s I n tro
du c to ry to th e M o dern H igh er A lgebra, an d t o C h a p X I I I o f
’
Th eo ry of E qu a tio n s
B rn s ide an d Pa n t o n
52 7
m
u
m
mh
f m
m
u
s
fu
u
m f
s
h y
u
u
m
u
h s
u
s
mt h d t h u gh t h ti lly mpl t
Th
t lw y
i t i p
ti
W h ll t h f
ly gi v
h mt
i
h
t
f
l
l
t
t
h
y
h
l
h
u
l
b i f
d
t
i
ll
t
t
g
p
mpl s m mth ds f limi ti n t h t
by
m p ti
lly u f ul
id
lim
th
i ti
u k w
L t us fi t
f
52 8
i
w
t
u
y
w
u
t
s
t
i
b
t
t
q
q
i
t
u
s
b
D
t th
0
d q5
d s u pp
q
y f( )
y t h qu t i s h v b
f mi
t h t if
du d t
i
wh i h f ( )
d <
t
t
l i t g l fu t i
f
1( ) p
i s v
th
tw fu
i h s im
u lt u ly t h mu t b
Si
m lu f wh i h ti fi s b th th giv qu ti ; h
y m
.
.
.
.
.
.
.
.
o
s,
o
c o n ve n
en
n
es e
t
e
os
r e
ex
e
se
c
n ce
so
e va
e
e
rs
ce
o
eo r
e
o
co
e e , a re n o
a
e re o re o n
e s
.
en e ra
an
,
S
o
n a
e e
e
n e c e s sa r
x
an
ese
e o
)
o
x
on
a
on
a
e e
,
er
co n s
o e
ee n
e
an
,
e
o
e
en o
ra c
ca
a
a
a re
s
e a
s ra e
en
a
a
o re
ra c
.
.
a
so
e
ex a
ca
on
an a
e o re
x
c
sa
=
on
a
re s e n
n ct o n
an
s
e
e
n a
on
o
on e
n
no
n
.
x
a
re
e
ra
an
e
e en
on a
s
n
re
e
an e o
o
e
o se
an
en
ce
ra
o a
n c
s
a
n
on s o
e re
e
or
on s
s
x
.
e
en c e
E LI M I N A TION
th e
mi
e li
fi
t
e x p re s e
t
n
o rde r
c o e f c ie n
u
si
S pp o s e t h a t
t e n o n e a t le a t
b e e q al t o z e ro
h
s s th
t h t th
n an
u
a
x = a, x
s
o
h
dit io n t
e
a t io n s
q
e co n
e
u
B
u
445
.
h t must h
my h v
a
,
a
a
a re
th e
f t h e q a n t it ies gt ( a ) ,
in an t is
e n c e t h e e li
P( ) P(B ) P( r)
I (B )
ro o
mm
ts
on
o
f
ro o
f( )
x
=
t
.
0,
mu t
s
,
0
a
f
ld b e t we e n t h e
e a co
( )
m
o
symm
v u
fu
Th e e x p re s s io n o n t h e le t is a
e t ric a l
n c t io n
o f th e
a t io n
ro o ts o f t h e e
x
a
n d it s
0
l
a
e c an b e
o
n d b
t
h
e
,
f( )
q
y
e t h o d e x pla in e d in trea t is e s o n t h e Th eo r
E
o
u
a
t
i
n
s
o
y f q
m
u
s
fu
.
m
m
sm
52 9 W e s h a ll n o w e x p la in t h re e ge n e ra l e t h o ds o f e li in a
ill b e s ffi c ie n t fo r o u r p rp o s e t o t ak e a i ple
t io n : it
ac
a e th e
ro c e s
l
b
u t it will b e s e e n
t
h
a t in
i
e
e
c
s
exa
,
p
p
i
a t io n
f
an
a
l
c ab le t o e
d
re e
O
e
pp
q
y
g
.
w
m
u
u
s
illu s t
Th e prin c iple
E le r
u
m
mi
E li
.
n ate x
dx
L et
tio n s ,
x
.
ra t e d
in t h e
f
llo
o
wi
n
an
3
w
b et
?
b
x
+
th e
een
ex
eq
+ d= o,
u
g
ex a
mpl
e
is du e t o
atio n s
f
x
2
+ gx + h : 0
.
+ k b e th e fac to r c o rre s p o n din g to th e
d
s
u
m bi
F mth
n
,
h
s
o
e t
p
p
e n
Eq
g
ro o
t
co
m
m
o n
h u
to b o t
at
uk w u i
u i w h v id
n
es e eq
ro
u
atin
no
n
q
at o n s,
g c o effi c ien ts
o
an
t ties
e
eq
a
.
a e
k w
f li
e
po
en
tic ally
ers o
f x , we
mb
h l + gm
—d
hm
gl +f
+
n
~
a lt
-
=
f 0,
c
—d = 0
f ,
cn
0
a
mh
o b tain
— a n + a — b = 0,
g
f
fl
,
s
d
k, l,
n
h s
.
E x a p le
an
u
.
m
u
Fro
t e s e lin e ar eq atio n s b y eli in atin g th e
we o b tain th e e ter in an t
d
m
—b
f
0
a
g
f
b
h
g
c
f
a h — ef
df
lo
h
d
o
ag
-
=0
.
uk w u
n
no
n
q
an
titie s l,
m
,
44 6
GH E R
HI
m
y
m
G
A L E B RA
.
u
=
c
an
x
0
0
<
)
x
n
s
i
o
a
t
h
e
e
f
t
n
t
o
i
n
a
li
,
Th e e
1 (’ )
( )
53 0
q
r s D ia l tic
e
l
e
s
t
b
S
an
t
i
n
r
t
e
a
d
e
a
s
d
r
s
s
e
e
i
l
e
x
y
b e e r e as
y
p
l
a
e
a
s
x
e
a
e
h
e
a
k
e
t
t
a
ll
s
e
W
p
h I eth o d of E li in a tio n
b e o re
v y
.
f
m
h
.
m
mi
E li
.
w
b et
n ate x
th e
een
eq
u
f
M ul tiply
c c es s io n
q
m
atio n s
x
s
yv
sm
.
E x a p le
u
u
f
an
tities
4
x
,
3
s
x
,
2
,
x
by
atio n
u
dd d
dx
4
bx
3
f
x
f
x
mi
n an
4
9
cx
+ d = 0,
dx
0,
2
b
x
+
+
0x
2
fx
e li
.
d wu
w wh h
v
x , an
3
ax
H e n c e th e
+ gx + h = 0
d th e s ec o n
ic
a e 5 e q atio n s b e t e en
is tin c t ariab le s
as
regar e
th e fi rs t e q
; we t
x
u
hu h v
2
2
2
x
g
3
h
gx
.
atio n
u
m
2
d
0,
0,
hx
?
3
x
by
x in
in ate th e 4
e c an eli
Th e e q atio n s are
eq
x an
0
luv
t is
0
a
b
c
d
a
b
c
d
0
0
O
f
g
h
0
f
g
h
0
f
g
h
0
0
mh
f w
Th e p rin c iple o f th e o llo in g et o d is du e t o B e z o u t
it h a s t h e a d an t a ge o f e x p re s s in g t h e re s lt a s a de t e r in a n t o f
lo e r o rde r t a n e it e r o f t h e de t e r in a n t s o b t a in e d b y t h e p re
c e din
t
all c o o s e t h e s a
xa
e
o
d
s
W
e
s
e
e
l
e
a s b e o re ,
g
p
’
an d
i
h
d
e o f c o n d c t in
t
e li
n a t io n
e
a
c
s
o
e
i
C
g
g
53 1
.
v
w
h
h
mh
h
v u hy m
m
E x a p le
.
mi
E li
n ate x
ax
Fro
mth
u
e s e eq
3
w
b et
+ b x + c x + d z 0,
we
h
an
(a h
d
Co
mb i
n in
h
gt
es e
-
two
bf)
u
eq
h= 0
gx + hx
+g
f) x
+ gx
2
fx
c
fx
2
a e
+b
2
atio n s
hv
ax
( ag
en c e
u
eq
2
atio n s ,
m m
m
u
th e
ee n
f
w
m
h
.
cx
+d
hx
x
2
+
(a h
f)
c
x
h
0,
+ (b h
atio n s
f
x
2
wi h
t
4 9 33
“
m
u
df: O,
.
f
.
44 8
HI
E
xa
l
m
p
2
e
.
m
E li i n ate
2
2
y +z
hv
We
by
a e
multiplyi
n
=
y
x,
ay z
,
m
pl
e
3
z
y
y
x
t
t
es e
(a
h
u
re e e q
+
2
2
33
x
n ate x ,
y
2
=
px
u
u
M ltiply in g th e fi rs t eq
2
en c e ,
hd u
b y th e t ir
eq
atio n
2
+
y
+
2
x
2
+ b +c
th e
e en
4=
~
u
eq
abc
by
x , an
2
atio n s
d th e
s eco n
x
3
3
3x y
w
p
4
.
m
E li i n ate
3
-
z
We
b tai n
-
3
31
q
=
w—
y) ;
(
3
-
2
e
o
2
(p
m
pl
b y y , we
,
q
xa
d
( ro u x) ;
3x y = p
1)
E
.
4 xy = qx + 19 31, w2 + y 2 =
qy ,
atio n
2
2
w
y b et
3
w
h
we o b tain ,
atio n s
2
.
2
mi
2
+ y = c xy
2
er
E li
.
x
atio n s
2
2
£
2+
xa
= bz x,
eq
_
a
E
u
th e
een
.
3
y
en ce
+x
2
G E BR A
8
+
h
2
z
w
b et
z
,
AL
z
h h
o
t
g get
GH E R
x,
y,
w
b et
z
e en
y
th e
5
y
x
2
z
)
u
eq
x
y
y
x
y)
2
atio n s
hv
a e
(y
-
z
)(
z
-
x
)
—
w
( y)
xyz
I f we
Sign o
h
fa
c
h
re
an
ge th e
mi
en ce
a n s
u
s ign
n altere
a
d
o
f
x,
th e
s ign s o
fb
an
d
;
—b — c
_
(y
(M
y)
xy z
mil
Si
arly ,
b
c a re c
c
a
x yz
c — a
—b
z
+
y
(
)
-
xy z
y)
h
an
ge
d wh
,
ile
E LI M I N A TI O N
449
.
a
2
2c
a
2
a
“
E XA M PLE S
mi
n at e
2
a=
x
mi
m
E lim
i
E li
n at e
m b
(m
x
n
y
et
w
a
f mth
r = 0,
+
g
0
1
ap
mi
n at e x
dx
r=
f mh
t
ro
2
2a
u
mm m
m
y
e eq
u
2
x
n at e
x,
'
u ti
a
+1
a
a
e e
,
2
+x
2
=
z
a ,
y
mi
E li
n at e
p
zx =
=
E li
n a te x ,
—
t
= a
,
x
t
ro
+y
mi
n at e x ,
y
= a
,
,
z, a
e eq
x
e eq
Z—
f
m
h
y
n ate x ,
E li
y
a
u
y
u
+9
u
f mh
t
b
mi
n at e x ,
x
H
.
H
.
A
.
y,
3
f mh
t
ro
z
3
+y +
z
3
b
m)
.
x
p9
3 —
x
,
z
x4
,
u
=
y
3=
3
c
.
u
4=
4
0
+y
.
atio n s
= oz
2
413
,
—
ons
2
e eq
e eq
3
5
1
(
.
atio n s
y
E li
0
atio n s
2=
ro
x = a,
(1
2=
e eq
2
=
atio n s
p
ro
mi
x
y,
y f mh
mi
x
t
3 ax
= c2
,
xy
,
m
h
f
g
93
E li
b
ro
,
an
e e
2
2
ons
,
ro
2
°
ons
O,
x = a
n a e x,
1
97
ro
n
,
atio n s
mf mth qu ti
—
m
1
m
+
+
(
)
y
y
E lim
i t
f
t
h
u
m
t
i
q
y
mi
E li
2
20
a
e eq
0
.
atio n s
y,
g
=
.
.
on s
eq
nx
ro
r
p , g,
m+ 1
.
n
th e
e en
2
a
n at e
E li
n
,
O
0
n x
n at e
b
0
.
x =
,
e e
ro
n
,
2 2 2
c
a
z
2
x
m
E lim
i
0
.
ons
a
e e
0
4
mf mth q u ti
0
m
m m
y+
y+
m f mth qu ti
ro
b
2 2
XXXI V
.
E li
b
4
2
+ du + a x ,
atio n s
2
+ 3/ + z
2
2
a
,
w
2
450
E li
13
“
3
a
+ +
z
y
14
m
i n at e x ,
x
z
m
in ate
.
n at e x ,
z
)
.
(
x
z
)(
y
19
.
t
e
20
.
2
S
.
hw h
e
mi
e li
is th e
n an
t
22
.
.
ax
)
f mth
—
ZN
u
at io n s
y)
e eq
(
(
2
4b z x,
2
u
ro
2
y)
x
u
e eq
ax
+ y
2
4o
2
xy
.
atio n s
—x
) (y
z
+
x
)
=
bz x ,
z
e eq
h
t
m
f
y
fl
abc
at io n s
2x + y
u
c
a
= a
+ a)
at io n s
xy
+ 6y
3x 3
c
3 3
60
by + z x =
b e,
n ate x ,
mpl y B
o
3
e eq
ro
z
y,
(
23
m
xy z
x
+y
2 202
5(1 6
3 b g=
of
mi
E
(
x
(
at
=
z
x
a +y
E li
e
atio n s
at
ax
21
u
= a
,
n at e x ,
t
x
z
ro
mi
E li
at i o n s
3
0
h
t
m
f
y
n at e x ,
hwh
eq
x
x
S
b
3
f mth
z
,
+x
(
u
z
+
(y
mi
E li
.
< >
z
ro
z
(
18
2
2 z
4a y ,
2
n ate x ,
+y
ro
th e
y
y,
mi
E li
f m
z
)
e
e eq
b
y
x
y,
x,
mi
E li
(y
17
y,
f
W
16
ro
2
a+ a+ i
E li
.
f mth
2
bx y
x
’
ez o u t s
my
2
xy
cz
ca,
xy z = a b a
= ab,
f m
ro
—
b
Pl
—
fiy
g
mth d t
o
e
3
( lg
/
0,
'
flll
f ‘
c
2
mi
o eli
’
a x
3
n at e x ,
b
'?
x
y
ro
f
y
'
2
c xg
/
m
dy
’
3
O
.
An
HI
d
53 8
D e n o t e t h e gi
v
x
f( )
u
=
f( )
Th e e q atio n
de n o te d b y a
x
t
h
e
en
qu
x
p0
AL
h
t
e
qf
equ a tio n
E very
.
GH E R
i
f( )
by
at o n
0 x
1
wh
0,
x
"
.
2
v
f
en
f( )
(
x
a
x
l
.
p
mgi
0 h a s a ro o t , re al o r i
i
x
s di is ib le b y x
( )
=
o re
e re
2
’“
m
ro o ts , a n d n o
degree h as n
th
n
G E BR A
y ; l t t h is
th t
e
a
so
a
b
e
n ar
a
)
fu
m
s
1 di e n sio n
n c t io n
of n
wh e re gt , (x ) is a rat io n al in t e gral
h a s a ro o t , re al o r i agin a r ; le t
A ga in , t h e e q at io n
a , s o t h at
t i b e de n o t e d b y a 2 ; t e n ct, ( x ) is di is ib le b y x
g
u
hs
h
¢ (
x
l
h
w
( )
qt,
e re
a ra t io n a
is
x
h us
v
5
x
9 2 ( ),
5
8
fu
l in t egra l
f( ) (
T
z
) (
90
a
x
)
.
h
Pro c ee din g in t is wa y , we
o
th e
en c e
wh
en
e
h as
x
u
q
an
f( )
at io n
y
v lu
f th e
o
a
es a
,
a
b t ain
,
)
“
m
fn
2 di
e n s io n s
.
in A rt 3 09 ,
as
ro o
,
y
w
a( )
.
(
n
a
,
s
t
o
)
a
0 h as
x
n c t io n
w
(
—
H
m
.
t
—a
a
o
)
.
s
v
f( )
s in c e
x
an
is
hs
e
3 ,
hv m h
v u
f m
u
f s
h
f
f m
h f
v
h
h
v
u
f
I th
b v i v tig t i
m f th qu titi
my b qu l ; i t h i s h w v w h ll u pp t h t t h
i
h
i
l
l
l
u
t
t
t
t
h
u
h
t
h
f
t
l
l
d
i
f
t
q
g
u
o re t an n ro o t s ; fo r if x h a s
A ls o t h e e q atio n c an n o t a e
a n t itie s a , a , a ,
an
f
re n t
ro
an
f
t
h
a ll
e
al e dif
o
e
,
y
y
q
2
s
z e ro , an d t e re o re
t h e a ct o r o n t h e rig t a re di fe ren t ro
x
an is
f
al e o f x
c an n o t
o r t at
( )
.
e a
n
e
a
e
e
a
es
n
c a e,
e o
o
s, a
ro o
n
o n so
a
s
n
as s
on
a
e
o
e
e r,
e
rela tio n s
.
e
s
s
a
e s e a re n o
o
To in vestiga te th e
53 9
co
eien ts i n a n y equ a tio n
i
efl
es a
an
l
a
2 ,
a
o se
b etween th e
a
a
e ren
a
.
s ,
ro o ts
an
,,
e
.
d
.
L et
u s
de n o t e t h e
x
an
d th e
n
.
x
ro o
+
pl
"
a
l
u
q
+
ts b y
n
.
e
+
a,
p2
b,
x
n
a t io n
p 2x
"
‘
by
2
+p
k
c,
—2
t
+
p
h
n
en
en ce,
w it h th
w 2 x
e no
u
+ 9
w
l
t at io n
+p2
m
n
Slw
n
o
a ve
y
ide n t ic a ll
—a
x
)(
x
—
b)
(
x
—c
)
f A rt 1 6 3, we h ave
.
—2
—l
h
_1 x + 19
(
h
we
O,
x
+
q
k
u
-
2
p
n
—l x
1)
S
n
—l x
(
x
—k
);
Y
TH E OR
Eq
u
i g
fi
c o e f c ie n
at n
S
p,
S
=
p2
su
= su
ts
w
ts ;
f lik e p o
o
m f th
m f th
t im;
m f th
t im
o
e ro o
o
e
U A TI ON S
OF E Q
p
e rs o
u
d
ro
ct
4 53
.
h
f x in t is ide n t it
s
o
f th e
ro o
t
st
y
,
ak e n
t wo
e
p3
S3
=
su
o
e
p
ro
u
d
ct
s
o
f th e
ro o
ts taken t
h
re e
e
I f th e
th e eq
co e
u
9
2 0,
zp
ffi i
u
ro
d
ct o
c en
t
o
at io n
b ec o
f th e
ro o
ts
f W is p a , t
m
h
.
en
h th
_
fl
pa
m
s atis fy
h
mth
th e
c
eq
u
b ic
u
u
e
th e
u
we
h
t
s ee
atio n
b,
at a ,
mth
fo r
Th e
are
e
e
x
If a , b , c are th e ro o ts o f th e
2
2
2
2
o s e ro o ts are a , b , C
e q atio n
e
re q
u wh
ui d u
eq
.
(x
(
a
o
—a
) (
2
cc
(x
u
= ab e
a
?
—b
)
(y
) (x
(
se
2
b
e
-
w
b
)
(
a
2
?
) (y
) (x
2
b
C
2
)
?
hu
s
th e
re q
a
u d u
ire
(
3
90
eq
) (y
c
?
)
O,
= O, if y = x 2 ;
2
3
=
0)
x + p 1 x + p 2x + 17 3 :
+ 101 06 + p 2w+ 19 3 )
2
(
“
an
d
2
if we rep lac e x
b y y , we
3
2
+
1
(
19 2 u
3
(
3
96
(
3
90
2
7
1
( 2
6
“
4
10106
(P2
o
.
is
atio n
2
2
9 1 90
+ p 2w—p 3)
2
0,
2
2P1 P3 )
0,
x
2
“
P3
2:
O;
b tain
?
h
)y
?
(
2
19 2
219
o
P0
f t
+ p 2x
)
(x
es
.
3
atio n x
)
2
"
—
—
=
x
x
a
x
x
x
b
x
c
( + )( + )( + )
p3
p 1 +p 2
T
al
.
e q atio n is
re
v u
th e
0
=
y
m
pl
)
p
atio n s
en c e
xa
e
—1
t3 — z t2 — y t — x =
E
v
— p3
,
p0
Ba b e
eq
atio n s
eq
.
,
v
te r
Po
f A rt 52 1 , we h a
22
So l
e se
o
9
2 ab
’
E x a p le 1
Fro
tat io n
e n o
mb y
h
9
Po
Po
d, wit
ea c
es
2
an
v
di idin g
on
1
10
3
y
)
-
p3
2
=0
wh h
ic
GH E R
HI
4 54
AL
mh u
u
um
fl
m
u
GEBR A
h
.
s s
Th e s t de n t ig t s pp o s e t at t h e re la tio n e t ab lish e d
se
d
r
o
o
l
an
i
o
o
e
t
l
h
a
b
n
e
l
e
i
l
w
o
d
r
t
c
e
a
i
n
e
d
in t h e p re c
y p p
g
r
b
e
h
n
a
l
t
o
t
e
i
s
e
a
t
i
n
l
o
h
r
e
t
e
r
o
f
n
b
e
h
e
r
t
n
f
o
e q a t io
q
;
h
e
i
n
o
t
t
a
i
s
h
t
t
t
ill
s
e
i
n
c
t
o
l
r
e
li
e
e
t
t
A
t
r
o
o
o f th e
n t itie s
a
h
f
e
o
t
1
an
n
n
a
t
e
li
i
w
e
e
se
o
s
r
o
f
ca e
q
y
pp
i
n
i
n
h
r
a
e
r
i
n
t
e
e
t
e
e
i
n
o
d
o
t
a
t
an
e
i
n
b
t
a
O
n
d
s
o
a
7c
a , b , e,
g
q
l
i
n
l
i
a
s
e
t
r
c
l
d
e
i
n
o
a
r
e
i
i
e
s
an
t
t
s
e
t
e
i
n
c
e
n
on e; t e
q
i
an
b
t
a
n
w
l
l
a
l
a
o
w
a
s
e
t
a
t
l
a
r
i
s
c
e
i
t
i
n
s
t
o
a
e
h
t
e
a
o
f
e c
,
q
e q a t io n is t h e re o re
e c o e fi c ie n t s ; t i
h
t
e sa
a in
at io n
e
g
q
su b
it h o e o n e o f t h e ro o t a , b , c ,
t h e o rigin al e q at io n
s tit t e d fo r x
54 0
.
u
s
s
.
u
h s
h
u
hv
h
u
u
us t
a
k e fo r
ex a
d le t
a,
b,
b e th e
c
th e
e
w
+
p1
x
a
u
y th
M lt ipl
es e e q
u
b
h
b
t h a t is
a
h hi
w ic
s
n al
i
g
th e
o ri
Th e ab o
a
li
a b le t o
c
pp
v
e
e
ro c e s
p
u
q
at io n
m ms
m
E x a p le 1
t at th e ro o ts
D
m
en o
f th e
o f th e ro o ts
So
.
ro
p
v
a
hu
s
es e
th e
,
p1
2
—
+
p2
a
p2
re e
e
p
e c ti
a
—
p37
9
O,
v ly
an
e
in t h e pla c e
Of x
u
eq
al es
eq
o
f th e
u
1
2
’
hs
u
an
d is
.
ro o
s
t
u
v
o
.
u
h vi
a
p ro gres sio n
h
.
m
— b2
we fi n d
2,
m
hv
2
—
2
°
a
4
2,
a:
h d
9
u
23
th e t ir
.
4x 3
atio n
etic al
3a
satis fy
ro o ts are —
t
.
u
m
us
6,
atio n
a dd ;
f an e q at io n a re c o n
53 9
re la t io n , t h e pro p e rt ie s p ro e d in A rt
t o o b t ain t h e c o p le t e s o l t io n
o re
h
z
d
is q it e ge n e ra l,
on
n a
an
o
res
13
a
e
o
du
h vu
t
1
a,
.
b , a , a + b ; t en th e su
ts b y a
o f th e
2
2
i
s
e i 3a
c ts o f th e ro o ts two at a t
b ;
2
2
( a b ) ; en ce we a e th e e q atio n s
is a
fi rs t
2
a
2 ,
103
ro o
3a
e
9
1
:
on
a
hm
e
a
2
p
u
w
i
h
i
t
t
q
s f limi t i
s f ydg
th e
arit
l
in
are
te th e
o
mth
p1
m
.
T
s
y
f
at io n
—
bc
by
qu
+ c
a
+
e
54 1
If t wo o r
n e c te d b
a n a s si n e d
y
g
W i ll s o
et i
e
e n a b le
s in c e
ymm
ys
'
u
m
en
60
a t io n s
s
e
+ ac +
—
fro
m
2
ts ; t
ro o
a +
su
h
sm
mpl
3
h
v v
h
hs
s
u
um
.
L et
an
u
f
w
u
s v
s u
hw h
w
u
m
m
,
(a
an
z
h
en
du
—b2
)
d fro
th e t
v
g gi
ts is 3 a ; th e
d th e p ro
ct
ro o
an
n
ree
mth
u
eq
e s e co n
atio n s
d
b
are
:
an
c o n s is ten
d
t
.
4 56
HI
mth
Fro
u
es e e q
atio n s
GH E R
3
2
Za b
37
Ea b
‘
(
e eq
or
3r ;
pq
.
5 é
,
v
So l
e
.
6
.
4x3
.
4 x3
8
gress io n
4
2x
.
u
O) 0a 2 a 2 :
4
a
.
+ b,
—b
a
—a+
,
— a —
b,
b
.
atio n s
+ 1 6x
2
+ 2 0x
2
9x
36
=
1 05= O, t wo
O, th e
two
2 3x +
52 x
24
— x 2 — 2 2x
3
2 4 x3
su
o
m f tw
o
o
f th e
O, t h e
ro o
ro o
ts b e in g 1
o
ro o ts
f th e
an d
t s b ein g in
.
b ein g z e ro
ro o ts
b e in g e q
7
.
ul
eo
g
a
.
mt i
e r c al
r
o
p
t wo
o
f th e
b ein g in th e
ro o ts
ratio
o
f
2
+ 4 6x + 9 x
o n e ro o
t b e in g
d ub l
o
e
an o t
h
f th e
er o
.
11
8x
o s 1te
in
pp
2
.
.
O
eq
.
.
10
ro o ts
2, o, 5
.
3x3
.
9
,
-
a
.
o s e ro o ts are
l 7 6x
5
4
x
7
3
th e
XXXV
.
°
2, 2,
.
h
w
atio n
3
2
3
u
b ed )
acd
a b cl
abc
E XA M PLE S
F mth
.
a e
—
pq
Z
GE BR A
hv
we
‘
AL
4
s ign
12
s
i
s
on
r
e
g
r
s
s
e
g
m
14
3
3 2x
.
.
4
x4
p ro gres s io n
mt
o th er
2
2 6x + 1 6
4 8x 2 + 2 2x
2 9 x3
3
7x
2 x3
2 1 x 2 + 22 x
eq
u lb
a
u
e r c al
p ro
t
O, th e
ro o
ts b ein g in ge o
mt i
O, th e
b ein g in
ro o t s
arit
hmti
e
cal
o
r
p
12
O, th e pro
du t
c
o
f t wo
o
f th e
ro o ts
4 9 4 73
r
o
p
O, t h e
ro o
r
e
s
s
i
o
n
g
2
52 0x + 1 9 2
.
ts b ein g in
O, th e
arit
hmti
e
c al
ro o
ts b ein g in
geo
.
o n e ro o
.
two
40
.
.
e ri c al
17
ts b ein g
.
15
16
ro o
.
6x
b ein g 2
f th e
.
.
n
o
.
3 9x
.
13
two
2x 3
t b e in g
h lf th
a
e su
m f th
o
e
TH E OR
If a, b,
18
th e alu e o f
v
.
If a, b,
.
th e
1
1
1
( 1)
19
c are
e are
th e
<
Fi
d th
e su
21
Fi
d th
e su
n
.
n
.
.
f th e
b)
m f th
o
o
U A TI ON S
eq
u
atio n x 3
2
px
~
O, fi n d
r=
+ qx
1
b
+
2 2
0
v lu
fi n d th e
a
e o
f
2
( )
e sq
u
e
are s an
r
o
o
wi th
1
-
d
m f th f u t h p w
o
4 57
.
f
2
equ a tio n
I n an
54 3
o c cu r i n p a irs
o
ts
ro o
—
20
ts
ro o
OF E Q
(2 )
1’
( 1)
Y
f th e
o
c
ub
f th e
e rs o
fi
rea l eo
e
i
es o
ro o ts o
ro o
ts
f
o
f
mgi
i
ts
e en
f th e
ro o ts
n ar
a
y
.
u s
f
i
S pp o e t h a t
O
s
x
( )
o s e t h a t it h a s an i
an d s
pp
ib is a ls o a ro o t
t at a
u
h
f
act o r o
(
f
,
mgi
u
q
wit h re a l c o effi c ie n t s ,
ro o t a
ib ; we S a ll s e w
i
at o n
y
n ar
a
f
f( )
c o rre s
x
x — a
—ib
)(
p
x
din g t o t h e s e t w o
on
ib ) ,
+
a
v
—a
m
h
y
:
Eq
h
u
a t in
t
o
g
h
th e
z e ro
re a l a n
2
d b by
hyp t h
o
e s is
is
n o
t
z e ro
R
en c e
f( )
x
is
en c e x
544
ti o n
is
x
a
)
b
+
2
.
h
f( )
en
x
s
a
ib is
a
s
l
o
— a
a ro o
.
t
yp
n ar
a
0
:
hyp t h
0 by
:
u
o
o
t ie n t b y
e s is
l
a so
a rt s ,
°
,
an
by
ib )
I n t h e p re c e din g
h
as a
a ir o f I
x)
0
p
(
ac t o r o f t h e e x re s sI On
p
f
f
a
x
e
z
mgi
di
0
y divi ib l
e x ac t l
(
h
h
t s is
z
Rb
H
(
or
2
ro o
+
h
L et
b
e di ide d b y
x
b
d
e
n
o
t
e
t
e
x
;
)
q
(
( )
'
a in de r, if a n
b
R
n
an d t h e re
x
t
e
R
,
;
y
y
a
ib , t
I n t is ide n t it p u t x
2
e
—
n
c
R
e
e
a + ib
b
0
a
x
+
;
(
)
(
)
an
h
.
Th e
Q
e
an
(
d R
(
x
’
0
.
a
)
2
b
e
,
t
h
at
is , b y
x
.
a rt ic le
mgi
we
y
n ar
a
f( )
x
h
ro o
ts
a
'ib
=
h t if th
th
(
t
a ve s e e n
,
a
en
x
e eq
a
)
2
u
a
b
2
4 58
Su pp
t
o se
f th e e q
a drat ic
q
o
u
h
N ow
e ac
u ti f (
f t s
on
a
AL
l= id
e=
,
e
l(
( )
h
o
x
x
)
O,
an
Y
2
5
“
x
h
f t es e
is a lwa
d t
.
at
W
mgi
th e i
a re
f(
( )
h
es e
m
v
fo r
re al
al
s
i
o
s
i
t
p
o
e
fo r
e
.
v
y
e ver
u es
m h
m
.
v
So l
th e
e
eq
u
atio n
4
6x
h
Also
6x
h
th e
o
s
th e
ot
— 1 3x 3 — 3 5x 2
4
er ro o ts are o
b tain e
m
o
f
st
ere r e
y—
th e
t s is
-
3
re q
o
as an o
N
,
t
3
an
at
e
u
q
a t io n
is , if a
g to th e fi rs t p air e
n g to th e s e co n
p air
ire
eq
atio n
2
x
h
wit
i
s
b
J
m
.
S
hw h
t
e
A
mgi
a
x
2
—a
n ary ro o ts
at
a
n
d
1
an
v
g gi
en
co rre
.
wh
it
e gree
ratio n al
.
—3
as
one
p air
o f ro o ts ,
an
d
air
.
a e
e
2
rt
2x
th e q a ratic fac to r x 2 2 J 2x
a e th e q a ratic fac to r
5,
5
.
is
+ 5)
4
x
E x a p le 3
h vi
0,
3
is als o a ro o t,
4x
ratic fac to r x 2
a
2 —N
er
n
(
i
f
.
x
f th e fo
2
—3
2+
a e
2
x
no
o
=
1
3
2
x
x
3
0
+
+
;
(
)(
)
(x
h as
e
fro
atio n
x
s
u
2 + , 3, 2
eq
o se ro o
N
d co rresp o n
hu
th e
or
.
C o rres p o n
T
s
d m
-
,
3,
an
l 3x
3
g g J
F m
u
u hd
wh
J
J
mu h v J J
/
/
h p
/
w h v
ddii
ud
d whv
ud
J
u d u
ro o ts are
E x a p le 2
c o ef
fi cien ts , o n e
H
}
re al val
fx
s h
t at
e th e q
or
hu
2
k w h
hv
ud
Sin ce 2 — ,J3 is a ro o t , we n o
sp o n din g t o t is p air o f ro o ts we
a
en ce
t
ro o
.
E x a p le 1
t at o n e ro o t is 2
h
s
u
u
s
y
9
a c o rs
y
n ar
a
i
t
h
ro d c t o f t h e
s
e
)
p
i agin ar ro o t ; t h e n
x
“:
f t i
ys p sitiv
fi
h
a
t
i
.
a
S e w t h at in
A S in A rt 54 3 we
y
e n t e r in
air ; t
c o e f c ie n t s , s rd ro o t
p
i
s a ls o a ro o t
ro o t t e n a
b
J
54 5
ra tio n a l
h
h
GE BR A
din g t o t
c o rre s p o n
ac o r
( )
<
1)
en c e
l=i b
a =
,
at
96
h
GH E R
HI
th e
B
x
2
u
2 5= O
.
atio n
2
0
—b
8x2 = 0,
2
2
+ x
eq
2
x
2
—c
.
h
u
I f p o s s ib le let p + i q b e a ro o t ; t en
—i
S b s titu te
p
q is also a ro o t
t e se al es fo r x an d s b trac t th e fi rs t res lt fro
th e s e c o n ; t
s
h v u
u
whi h im
c
is
po ss ib le
u
n
les s q : 0
.
u
m
d hu
.
4 60
H
HI
we
en c e
s ee
m u y
y mi l ;
( )
b igu it ie s
i
ii
( )
a re
u
n
a c h an
us t
u
s
G E BR A
.
ct
each
co n
t in
u
a t io n
f Sign in t h e
o
a
Sign
th e
AL
th a t in t h e p ro d
b ig it repla c e
a
i
an
()
o rigin al p o l n o
11
GH E R
sb f
e o re
an
d
a
lik e ;
o
e
g
ft
er an
u
f Sign is in t ro d
mt u f
u
a
mb iguity
c e d at
th e
en
o r s et
o
f
a
m
d
.
h
f m
m h h
u
le c a s e an d s pp o s e t at all
t h e a b ig it ie a re re pla c e d b y c o n t in at io n s ; ro
( ii) w e s ee
b e r o f c a n ge s o f S ign will b e t h e s a e w et e r we
t a t th e n
e r S i n s ; le t
k
e r o r t h e lo
t
a
e
t
h
e
r
h
t a e th e
e
t
g
pp
pp ;
b e r o f c an ge o f Sign c an n o t b e le s t h an in
th e n
L et
h
m u
ak e
s
um
k
u
um
th e
os
n
a vo
ra b
u
h
w
s
h
us
s
—
an
hs
wi t
h
fi
sm
s
f s ign is t h e a e a s in t h e
a ddit io n al c an ge o f S ign at t h e e n d
s e ri e s o
dt i
an
h
h
upp s
us
u
y mi l
o rigin al
l
o
n
o
p
a
.
f
s
s
y mu
v
v
I f t en w e s
o e t h e a c t o r c o rre p o n din g t o t h e n e a t i e
g
l
t
i
li
an d i
a in a r
ro o t s t o b e a lre a d
d
o g e t e r, e a c h
t
e
act o r
g
p
a c o rre s o n din
o
a
o
s it i e ro o t in t ro
t
x
c
e
a
t le a st o n e
p
g
p
c an
e o f Si n ;
t
e re o re n o e
a t io n
c
a
n
h
a
v
e
o
r
o
i
i
e
t
v
e
g
g
q
p
ro o t
t an it h as c h an ge s o f s ign
m
h
y
h f
s h
h
du s
f
s
m
u
.
s
u
f
u
s
v
A gain , th e ro o t o f t h e e q atio n
x = 0 a re e
al t o t h o e
)
q
b
ut o
of
0
o s it e t o t e
x
i
i
n
S
n
t
re o re t h e n e at i e
e
;
pp
)
g
(
g
= O
ro o t s o f
x = 0 a re t h e
o s it i e ro o t s Of
x
b
u
t
t
h
e
p
( )
;
)
n
b e r o f t e e p o s iti e ro o t s c an n o t e x c e e d t h e n
b er o f
c an
s o f s i n in
e
x
t
a
t
i
s
h
t
e n
b e r o f n ega t ive ro o t s
;
,
g
g
)
o f f x = 0 c an n o t e x c e e d t h e n
b
r
e
o f c an e s
o
S
i
f
( )
g
gn in
f
f
um
h
f
96
)
m
E x a p le
hs
f
d
C o n si
.
hm
v
v
h
w
h
h
v
f
)
h
v
mu h v
h
H ere t ere
p o s iti e ro o ts
are
t
c
o
f
um
um
u
th e
er
h f
eq
atio n x 9
an ge s
o
f
s ign
5x8
,
h
t
x
o
t
f
S i gn
,
x
erefo re
t
erefo re
it
st
uu
\
.
v
1
2
3
.
.
.
e
th e
3
4
x
4
6x
x
4
+
eq
u
h
hh
.
.
ere
at n io s t
are
et
ere ar
hv
.
XXXV b
.
3
1 3x —
x
3 5x 2
2
+ 5x + 2 x
o n e ro o
.
t b ein g
o n e ro o
o n e ro o t
t b e in g 2
b e in g
ree c
h
an
two
ges
re e n egati e ro o ts an
,
atio n s
1 0x 3 + 4 x 2
3
4x
m
m
E XA M PLE S
So l
h
0
2
t
— x 9 l 5x 8 + x 3 7 x + 2 an d ere t
,
th e gi en e q atio n h as at
o st t
a e at leas t fo r i
agin ary ro o ts
- -
h
7x
erefo re
.
A gain
3
um
— 3
2
d
TH E O R
4
3
4x
“
x
.
6x
v
h
5 So l e th e
J 3 a n d an o t e r 1
.
F mth
or
one o
f
6
.
8
.
wh
F mth
F mth
or
or
.
.
ts
one o
m
n
.
t
ro o ts
2
x5
4
x
es t
di
e e
e
u i wh
wh
qu i
i
u
f
q
at o n
at
Fi
16
ati o n
x
F
.
9
in
d
d
l5
0,
ro o
o ne
t b e in g
wi h
e n s io n s
fi c ie n
rat io n al c o e f
t
at o n
J
—
.
o
th e
ts is
J2 +
at o n
e
o s e ro o
th e
eq
ts
f th e
u
ro o
t
3+
ts
5
-
J 2,
1
l
: :
1
,
t
o
e
_
2
4
J
J
.
3
.
at o n
O
=
.
le as t fo
at
a
in fe rred
e
10 —
res p e c t in
6
4x +
th e
co n
d
itio n t
u
1
( )
t wo
ro o
ts
( 2)
th e
ro o
ts in ge o
eq
al
h
hm
.
m
.
ro o
ts
o
f th e
(1
If a , b ,
al e o f
v u
u
r
umb
er
n
o
mt i
p
x
p
u
hw h
hw h
qu i
ro
ts
u
f th e
eq
n ar
ro o
o
2
mgi
f i
o
a
r=
x
q
f o pp o s ite
e r c al
ro o
at io n
3 = O?
2x
.
3 —
x
at
but
4—
v
g th e
s ign
i
n
es
s
o
r
g
y
my h
O
a
a
v
ts
o
;
.
c a re
th e
ro o
-
ts
a
o
at o n x
e
)
.
—
1
(
f th e
eq
u
23
ts
o
f
+p
7
.
0
=
a re
1,
a,
B,
s
are
hy
hw h
e
e
a re
t
—
3
x
p
x
2
x — r=
3
a
25
.
at
q
0, fi n d t h e
( b + c) ( c + a )
2a b
2
.
in
)
a tio n
21
ro o
1
“
f th e
e
4
3
2
=
I
f
t
h
r
t
o
e
o
s
o
f
t
h
e
a
t
i
o
n
x
x
x
r
x
s
18
0
eq
+q +
+p
3
e t ic al
arit
an d if t
t at p
p ro gre s s io n , s e
in geo etric al p ro gres s io n , s e t at p gs = r2
19
co
.
— x 4 + 4x 3 — 5_—_O h as
7
1
rat io n al
t
re e
e e
—
4
1
2
x
5
x
+
+
2x
,
h h d g wi h
J
J
i
f h
u
q
e ig
2
atio n
1
a re
J3
4
are 4
ts
o s e ro o
t h e leas t p o s s ib le
I f th e
ts ,
.
at
n
9x
m
o s e ro o
3 x4
e
8x
r ]
J
t b e in g
2
46 1
.
.
x
17
o n e ro o
7
h my b
.
u
w
f lo
e n at u re o
hw h
W
15
eq
0,
UA TI O N S
.
is
wh
f
d th
Fi
.
S
14
a in ar
g
y
i
at io n
o
e eq
mth
Fo r
12
13
at io n
ts
o s e ro o
u
1
J
.
ef
fi c ie n
u
5
4x
eq
OF E Q
f
3
J + J 2f
10
11
e eq
2
Y
x
q
2
rx
“
a
E
.
.
s
0, fi n d th e
v lu
a
e
462
HI
To fi n d th e
54 8
in tegra lfu n c tio n of x
GH E R
AL
f
f (
x
"
2
va lu e o
.
G E BR A
.
f ( x ) is
h ) , wh en
l
ra ti o n a
a
.
L et
f( )
=
x
x
p0
"
'
p,
h
hv
E x pan din g e a c
a
w
rs o f h , w e
e
o
p
po
"
x
+ p lx
"'
l
"
p2
x
(
h
h
re
sult i
d
an
+
1 ) p 1x
n
s
Q
-
"
(
+
fu
(
2
-
— l
n
y w itt
a ll
h
st
w
u
’”
en
r
es
eac
n
s ee
t
e
a
o
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e
t
a
)p
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f (w
to
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=
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n
h
f(
e
h ) is
f
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th e
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rs t,
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en
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n c
e
o
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x +
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on
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a
a n
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m t f t h D iff ti l C l
i
f
y
i
h
l
f
p
(
)
m; t h fu ti s f ( )
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t
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bt i f ( ) f m
w
m
ultiply
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a
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er
'
t iat io n
e re n
’
f h , we h a
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s
v
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an
en
h
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,
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e vi
t a in
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n
.
c c es s
f<
>
fu
{i f
b
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t
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e le
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h in t h e pla c e
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n c t io n
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din g
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f
B y writ in g
a s c en
n
3
"
a re
en
s
on :
e
s
in
p
'
or
eo re
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e
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ult
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H
o
a
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n
re s
3
’
a
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or
s
—2
n
?
t wh o k n o ws t h e
case
a
x
2 ) p gx
n
"
'
ud
(
IL)
n
in t h e
h t th b v
f T yl
Th
w
b
i
tt
t
h
f
m
y
f ( )
ul f diff i ti t hu t
h t mi f ( ) b y th i d
i i h th i d
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dim
il ly b y
u iv di fi
Sim
lu s
ill
a rt ic la r
p
cu
p
)(
en
r
n c t io n s f ( x ,
d th e
)
derived fu n c tio n
s ec o n d, th ird,
Th e
h
i
n
t
e
g g
a rran
x +
p
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an
th en
+ p n _, x
e
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s
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+ p n —1 x
m
te r
n
T is
pz
2
"
+
x
s
f (h )
e r c al
w it h sp
re
ect
464
HI
E xa
H
m
l
p e
e re
d
Fin
.
th e
dvd u
we i i
e s
6
u
o
v
by
GE B R A
AL
gin g x in to
an
1
2x + 5x
2
3
x
.
x
3 in th e
Or
m
e x p re s s io n
.
.
—1
5
39
15
h
fc
3
4
—
2x
x
lt
c c e s s i ely
—2
—1
2
re s
GH E R
1 32
_
2
1
_
2
11
2
17
2
23
s
5
2
_1
13
5
2
1 8 2 = q3
hu
b rie fly t
o re
97
2
H
my h
It
w k
or
th e
en c e
a
u
lt is 2 x
m k d th
e re
ar e
23x3
9 7x
’
H
at
2
o rn er s
131
1 8 2x
pro ces s
is
c
Co
.
h
iefly
mp
A rt 54 8
are
.
u ul
sef
in
nu
.
mi
er ca l
.
550
fu n ctio n
.
f
L et
We
res
4
If th e varia b le
x
l
ch a n e
w
i
l
g
( )
d
0 an
hv
a
c +
h be
an
ha n ges co n tin
c o n tin u o u s ly fro
x
c
mf (
v
w
t
o
y
ue
al
f
u o u s ly
a
)
ro
to
m
f (b )
to b th e
a
.
y
f x l in g b e t we e n
s o
db
a an
.
e
h
»
w
k
) f<
fl
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e
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m
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e
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,
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c
c
h
f
s h
sm h
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an d
h
c
d b y t a kin g h s all e n o g t h e di fe re n c e b et we en
)
(
e
a ll c a n
en ce to a s
c an b e
a de a s s
a ll a s we
l
a e;
c
e
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p
( )
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e in t h e
a ll c a n
in t h e v a riab le x t e re c o rre sp o n ds a
g
n c
ro
a t o b , th e
x , an d t e re o re a s x c h a n
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( )
an
f
m
f
h f
h s
f
h
m
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m
I t is i p o rta n t t o
551
a lwa s in c re a s e s
ro
x
( )
b
u t t at it
to
b
as se s
,
p
( )
an
s dde n c an
e
s
o
t
i
e
y
g ;
a
b
i
t i es it
e
d
ec
r
e
as
n
y
g
.
f
f
m
u
y
m
n o
f mf ( ) t f ( b )
f m v lu
m ms it my b i
h
h
o
ro
,
on e
e
a
e
a
v
th e
s
If f ( a)
equ a tio n f x
( )
A
ra d
g
.
w
u
a
an
ro
v
ed
t h at
sf m
h
h
ro
a
de c re a s e
f( )
e t o t h e o t e r wit o t
a n d a t o th e r
n c re a in
g
or
s
m
u
an
0
u
d f(b )
m
u st
m
f
a re o
w
co n
lie b etween
tra ry
a an
h g g du lly f m t b
lly f m
b
h
f
t
d
t
f( ) f( )
S x c
t p ro
fu
u
.
u
552
fu
e no
Th e s t de n t wh o h a s a kn o le dge o f t h e e le
t ra c in g will in an y pa rt ic la r ex a ple fi n d it e a s
ra d a l c h an e
o
f
v
al e o f
g
g
f ( x ) b y dra in g t h e c
u
m h
f
t ic e t h at we h a
a
f
es
ra
a
ro
a
o
,
an
a
o
,
db
th e
e re o re
s ign s
t
en
s
f C u rve
o llo w t h e
o
s
yt f
u v y
o
f ( x)
r e
th en
o n e ro o t
Qf
.
fu
n c t io n
mu t
s
p
f( )
ass
x
t
h
c
h
an
h
ro u
g
s
e
g
a ll
TH E OR
mdi
Y
OF E Q
UA TI ON S
465
.
v lu ; b t si f( ) d f( b ) h v t y ig
mu t li b t w t h m; t h t i ( ) 0 f m
lu
th
db
v lu f b t w
ly
w
t f ll w t h t f ( )
0 h
It d s
b
it f ll w h t if f( )
i h
d
d f (b ) h v t h
d b;
m
t b
w
db
ig f( )
0h
in t e r
ate
e
e va
e o
a
x
oe
S
n
e en
n o
o
t
n e
an
e
a an
o
a
o
o
et
ro o
as n o
x
e
s,
a
x
=
t
a
ee n
as o n
an
a
a an
u
u
b
f( )
on
n c
s
x
e
s t it
ut
e
fo r
t
et
e
e en
a
e sa
e
at
x
th e
lea s t
o n e r ea l
.
u
va l e s
0,
co
co
en
,
+
00
v
— 00
f(
:
I f p ” is p o s iti e , t h e n f ( x )
is n e gat i e
oo
a n d if p
co
an d
554
ns
.
m
fu ti
v y th
S
o r so
a
.
I n th e
c c e s s i el
ra r
=
x
o n e ro o
553
E very equ a tio n of a n o dd degree h a s
ro o t who s e s ign i s o pp o s i te to th a t of i ts la s t ter
s
e co n
a
.
o es
er
an
a
e en
e
e
s
e z e ro
n ce
u
es
a
v
=
f
y
.
w
0 h a s a ro o t l in g b et e e n 0 a n d
x = 0 h a s a ro o t l in
b
e
t
e
e
n
0
g
( )
y
w
E very equ a tio n wh ich i s of a n even degree a n d h a s i ts
n ega tive h a s a t lea s t two rea l ro o ts , o n e p o s i tive a n d o n e
m
.
la s t ter
n ega tive
.
Fo r in
s
t h is
ca e
+
is
but
an d
at i
n e
g
an d
co
00
v;h
e
a ro o
.
en c e
y
0
f( )
=
f( )
0 h as a
ee n 0 an d
x
t l in g b e t
w
f(
p
=
—
)
w
ro o
+
00
;
y
t l in g b e t
w
w
e en
0
f
555 If th e exp ressi o n s f ( a) a n d ( b ) h a ve c o n trary s ign s ,
0 wi ll lie b etween a a n d b a n d
a n o dd n u
b er of ro o ts of ( x )
a
f
h
s
a
i
n
i
t
h
n
r
a
b
v
e
t
e
e
s
e
e
o
r
o
o
r
n
u
e
n
d
h
a
t
o
a
e
v
n
n
b
r
e
,
(
g
if ( )
)
ll
w
een a a n d b
o
t
s
w
i
li
e
e
t
b
r
o
o
f
m
.
f
f
m
m
.
u
s
t h at
h
h
is gre at e r t a n b , a n d t at a , ,8 ,
= 0
w
re re s e n t a ll t h e ro o t
f
i
n d 6
i
o
x
h
l
b
e
n
a
t
c
e
e
e
a
f( )
p
L et ¢ (x) b e t h e q o t ien t
en
x
i
s di i de d b y t h e p ro d c t
( )
—K
x —
x —
x —a
x
t
n
e
);
) ( B) ( y ) (
(
—a
iv — 3
w
ff —
6 73
S pp o
e
a
s
u
wh
)(
H
f( )
b
f( )
en c e
a
=
=
(
b
(
a
—
a
—
a
w
H
.
H
.
A
.
v
h
u
)(
7)
a
a —
(
)(
y)
—
b
b
Bfl
)(
7)
(
09 6)
K S a
a
)q ( )
(
1
) — K
M “)
(
mu t b
m
1
-
N ow d
an d ( b b
) a
( )
( )
ro o t o f t h e e
a t io n
q
lie b e t e e n a a n d b [A rt
u
f
wh
s
.
h
.
f t h e s a e S ign , fo r o t e rwis e a
x
o ld
0
a n d t e re o re o f
,
( )
o
ic is c o n t ra r t o t h e
p
e o
h f
wh h
f
y
:
wu
hy
30
466
t
h
p
GH E R
HI
es is
H
.
if f ( a )
en c e
an
f
d
AL
b
( )
G E BR A
ha
v
.
y
si
t ra r
co n
e
n s,
g
th e
ex
re ss i o n s
(
b
(
a
mu t h
v
_
y
a
)(
a
)
—
a
B) (
(
m)
a
—
(
b
(
y)
a
y
f
)
—K
)
K
,
s
fi
A ls o t h e a c t o r in t h e rs t e x p re ss io n
h
a c t o rs in
t
e s e c o n d a re a ll n e a t i e ;
a re a ll
o s it i e , a n d t h e
g
p
s t b e o dd, t a t is t h e n
b e r o f a c t o rs
b er o f
en c e t h e n
a
K
s t b e O dd
ro o t
, ,3 , y ,
b er o f
Si ila rl if ( a ) a n d ( b ) a e t h e a e S ign t h e n
I n t is c a s e t h e gi e n c o n dit io n is s at is e d
ac to r
st b e e e n
if a , ,8 , y ,
K a re a ll gre a t e r t h an a , o r le s t an b ; t
s it do es
x
a s a ro o t b e t
o llo w t at
0
h
e e n a an d b
n o t n e c e s s a ril
( )
s
h
a
e co n
v
t rar
um
s
s ign s
mu
m y f
s mu
v
f
I f a, b,
.
f( )
H
If r
o
u
e re
th e q
ft
e
hm
h
.
u
f( )
h
=
hv
a
O
s
a
i
n
n
f( )
g
a dis t in c t ro o t
x
al
a re
th e
s
a
-
to
)(
b,
a,
a, s
P. (
it is
c as e
f
er
t itie
an
h
os
z
a re e
q
t is
hv
f
96
In
mu
.
c,
x
f
f
yf
556
.
90
a
c
) (
um
(
)
a re
n o
bl
(
hu
w
f( )
a t io n
se
—k
)
t
h
fi
.
0, t
x
h
en
.
yu
n e c e s s a ril
t
c,
”
90
sm
v
s h
c
-
to b, t to
”
um
u
(
h
h
f th e eq
—l
a
c
a
e,
a
v
ro o t s o
v
n e
u
q
al
.
en
90
u
ie n t s t ill t o s p e ak o f t h e e q at io n
eac
o f the e
a l ro o t b ein
c o n s ide re d
q
g
co n
ro o t s ,
en
h
u
s
.
557
If th e equ a tio n f(x ) = 0 h a s r ro o ts equ a l to
'
equ a tio n f x
ll
0
w
i
h
v
r
a
e
1
r
t
a
o
o
s
e
u
a
l
o
t
( )
q
.
a,
th en
th e
.
t
h
¢( )
L et
x
f( ) (
en
x
W rit e
x
b e th e q
u
o
tie n t wh e n
f( )
x
is divide d b y
x
h in t h e plac e
o
fx; t
hu
s
—a
g
—
(
f (x )
h) ;
+
x
hs
I n t i ide n t it
y
,
by
eq
u
at in
t
h
e
g
‘
c o efli c ie n
f( )
hu s f (
T
’
th e
e
x
)
co n
u at io n
q
t ain
s th f
e
f( )
’
ts
o
f h , we
w
(
x
i s,
(
x
x
a c to r x
0 h as
r
1
a
re
ro o
ts
p e at e d r
e
u
q
l to
a
hv
a
m;
— l ti
a
.
es
e
t h at
HI
468
GH E R
AL
G EB R A
.
u
h
m y
v
ut
B
1
r
a
s
h
0
x
n
i
o
t
a
e
h
e
t
a
t
f( )
e q al
q
,
a t io n
e
e
h
t
c
e
n
e
x ,
q
f
o
n
"
o
i
t
n
c
d
f
(
)
rs t de ri e
i
s th e
x
n
i
o
f ( )
t
a
e
e
h
t
l
r
a
l
i
i
s
a
q
o
t
l
a
e
t
s
o
o
r
2
;
a e r
q
st
0
x
f ( )
e
s
e
T
11
0
O
S
d
n
a
a
t
o
l
a
—
e
s
t
o
;
3 ro
a e r
q
st
al
e
e
h
t
r
e
o
sc
i
d
t
o
q
s
u
l
e
b
a
n
e
s
e
i
t
e
s
o
i
ll
s
n
i
o
a
t
r
i
d
e
co n s
5
9
5
r
t
f
A
o
d
o
h
t
e
e
h
t
n
a
t
l
e
b
r
o
t
s
es
l
i
t
O
x
o
f
s
t
ro o
u
fu
fi
v
uu
mu h v
mu h v
w mm
w
h
u
h
f( )
o
u
If
56 1
ro ve th a t
p
.
a,
b,
th e
a re
c,
’
Bu
h
f(
t
'
e n c e f ( ) is
b er o f ( 1 )
e
f( )
x
(
:
x
—b
u
eq
0
m
pl
e.
I f Sh
f
th e
ro o ts o
al e o
c o e fli c ie n
th e
to
(
)
t
h
d8
5
-
4
su
px
)(
x
—c
m f th
o
‘
qx
e
th
k
po
—k+ h
)
th e
(
)
t : 0,
2
ar ex p re ss io n s
h d
ol
5x
4
4p x
3
2 gx
.
fo r
f (x )
f (x )
x —
x — c
b
f (x )
9
x —
d
f (x )
’
e n ab
ro o
w
.
f (x )
mil
0, t
:
h
rig
t h an
-
.
x
’
d si
f( x )
in A rt 1 6 3,
—a
te th e
an
en
an
x
f h in
o
L et
t
.
.
x
—h
)
+
fl )
en o
f S4 , S 6
u
“
x
vu
h
.
f( ) fl
ati o n
fi n d th e
u
i
u
d
.
u
equ a t o n
re s lt o f t h e p rec e din g a rtic le
o f a n a s s ign e d p o we r o f t h e
m
.
xa
)(
—
90
a
x
e re o re , a s
x
Th e
56 2
fi n d th e s u
E
a
f ( )
t h at is ,
to
t
=
ul
h f
eq
x
mm
h)
to
m
(
x +
al
ro o t s e q
’
a; _ e
le
ts
ers o
s us v y s
i
f
u
q
o
an
f th e
i
er
ea
e
at o n
ro o ts o
.
f th e
Y
TH E O R
H
en ce
4
5x
by
3
4
+ px
By
eq
u
a
U A TI O N S
OF E Q
469
.
dd
itio n ,
2 qx = 5x4 + ( 8 1 + 5p )
x3
+ (8 2 + p S l )
x
2
( S 3 + P3 2
fi c ien
atin g c o e f
5g)
( S 4 + 17 3 3 4 9 3 1 )
x
"
ts ,
wh
wh
wh
wh
h v u
en c e
3 2 + Ps l z 0,
To fi n d th e
u
v u
al e o
f S , fo r o t
v
M ltiplyin g th e gi
u
en eq
+p
res
wh
wh
u
u
S b s tit tin g fo r
lts , we o b tain
u
hu
Pu t k = 5; t
x
in s
Sh
u
t
a
c
ce
.
f h , we p ro cee
pu t
th e
qS k_3
— 5
p
fo llo
v u
al es a ,
5t
b,
c,
d,
24
2
e an
d
.
5g
3 g2
6p 3 q
u
h
W en t h e c o effi c ien t s
56 3
in g ex a ple
c ee d a s i n t h e
o llo
6p t
wh
wh
wh
; t en
en c e
S _I
e n ce
5
en ce
a re
n
th e
su
o
x
f th e fo
3
f( )
x
f (x)
’
8
4
t
2
r c al
4p
t
we
my
a
.
rt
ers o
po
= x3 — 2 x2 + x
3x
-
um
e i
2 x2 + x
2;
2
-
f w
m
m F d
m
u h w
in
.
h
c ces s io n
o,
ts _2 z
s
.
+ fin
5
p q
s
w
k
2
0,
.
as
,
+ gee
k = 4 , 3 , 2 , 1 in
S3 + p S 2
E x a p le
d
5
-
k—3
6
=
Se p
.
3g;
s
en ce
To fi n d
4
=
S 4 p + 4p q
cc es s io n
S5
hu
1
en ce
p
s
en ce
Pu t k = 6 ; t
-
3 _
S3
k
by
—_
en ce
al es o
er
atio n
"
x
en ce
4x + 1
3
27
1,
.
f th e
ro o
ts
o
f
a
dd
in g
47 0
HI
GH E R
G
A L E BR A
.
4
1
10
H
th e q
en c e
u
o
2
3
ti en t
5
2
+
is
5
E XA M PLE S
1
.
3
.
4
I f f( x )
4
If
.
2
f( )
= x4
If f (x )
f( )
If f( )
= x4
x
XXXV
.
7 6x
3
2
1 x
2x
4
l 3x
1 7x
s
2
6 4x
v lu
a
v lu
a
b et
.
een
an
t
3
2
.
e
t
3
e en
a ro o
e
o
t
t
t
e
o
t
e
.
x
12
.
5
x —
.
5
x — v3 +
an
at
t
4
at
et
one
e
x5
5x
e en
o
n
4
an
2
f(
e of
x
2
+
x
v lu
a
+
x
x
x
+
e o
h)
f
x
(
f
f(
x
h)
a
ro o
0 h as
+ 6
+ 3x
3
2
h as
3 5x
.
t
a ro o t
.
1 2x 2
1 2x
e en 2 an
3
3
=
0 h as
a
ro o
4x
2
1 9x
2
0
as a ro o
t b et
w
een
2
an
.
c
14
.
3 x5 + 6 x 3 — 3 x 2
16
.
6 —
33
2 x5 — 4x4 + 1 2 x3 — 3 x 2
4
— a
(
a
t
.
a
.
x
e e
4 —
a
ts
ro o
3
6 x + 1 2x 2
1 3 x4 + 6 7 x3 — 1 7 1 x 2 + 2 1 6 x
.
x
an
x4
11
15
.
e o
5
at o n s
e
3
5x
et
er
2 0x
4
2
at o n
e
an o t
an
x
e en
5—
x
17
1
at o n
e
4
10
13
at
e
atio n
eq
e o
.
an
e
.
th e
at
1
e
een
an
w t
an
.
be
e
a
ff ( x
e o
h h
u
7
w 0 d
h
Sh w h
7
+
qu i
tw
d
d
b w
d
Sh w h
8
h
qu i
tw
d
d
h b w
d
Sh w h
9
h
d
b tw
4
d
S lv
h f ll wi g q u i
u
wh i h h v
q l
6
S
v lu ff(
lu
ff (
1 2 9 , fi n d th e
d, fin d th e
cx
.
6 5, fi n d th e
1 9 , fi n d th e
b x5
c
.
9 x + 7 , fi n d th e va
2
2
7 2x
ax
x
.
3,
x
=
If
.
5
be
a
10
.
8 x4 + 4 x 3 — 1 8x 2 + 1 1 x — 2 =- 0
.
— a 3b
z
0
.
d
G
H I H ER
I f t h e p ro p o s e d
po
t
h
it is
en
as
1
9
i
l
+p2
n
t t
—l
h
at
re/
+
th e
"
GE BR A
"
2
x
re
+
i
u
q
u
re d e
q
p
1
n
p
x
0,
z:
a
w ill b
a t io n
e
y
2
f m
h
.
is
at o n
"
en
9
2
“
u
q
-
v id
e
"
x
“
e
AL
We
0,
.
u
t h e o rigin al e q at io n b y c h a n gin g th e
w ic h is o b t ain e d ro
b egin n i n g wi th th e s eco n d
s ign of every a ltern a te ter
m
.
m
.
who s e
ltip li ed b y
an o th er
To tra n sfo r a n equ a tio n in to
56 6
equ a l to th o se of th e p rop o s ed equ a ti o n
u an ti ty
q
m
u
ro o ts a re
i
v
n
e
g
a
.
f (x )
L et
i
g
e
f
v
u
q
en
u
q
at io n
0 b e t h e p ro po
an
tit
is
f
h f
y
Pu t y
.
=
q
s
m
E xa p le
s
mv
Re
.
o
e
an d
x
u
q
h
a t io n
at
a
fractio n al
multiply
eac
c o ef
fi cien ts
3
§
1
x
3
-
fro
all
en
th e
c le a r a n
e
re
u
q
u
i
q
re
d
at io n
h m
hu
b y q3 ; t
ter
i
mth
e eq
s
atio n
3
2
s
3:
0
m b mi
th e ter
u
—0
2
u
h
.
8
3
By p ttin g q = 4
we o b tain
is t o
on
d let q den o t e t h e
t
x
f mti
or
an
.
—
Pu t
t
SO
x,
Th e c ie u s e o f t h i t ran s
rac t io n al c o e
c ie n t
ffi
s ed e
ec o
e
n tegral, an
d
on
divid
in g b y 2 ,
2
—
3y
y
3
m
56 7
To tra n sfo r a n equ a ti o n i n to a n o th er wh o s e
reci ro ca ls o
p
f th e ro o ts of th e p ro p o sed equ a tio n
.
th e
ro o ts a re
.
f( )
L et
1
0 b
x
t h e p ro p o s e d
e
u
q
i
at o n
; p t y
u
so
u
u
d
t
q
q
9
O
f th
h i f u f t h i t sf mti i t
v lu f p i s wh i h i v lv ymmt i l fu
t
v p w s f th ts
g
t
h
e
en
n e o
a
es
o
ne
a i
e
ex
o
th e
re
e c
e
i re
e
se s o
re s s o n
er
o
a io n
s
c
e ro o
n
.
h
at
is
ran
o
t
e
or
s
a
on
s
e r ca
o
Ob
t ain t h e
n c t io n
s
o
f
TR A N SFO R M A TI ON
m
E x a p le 1
I f a, b ,
.
x
fi n d th e val u e
th e
c are
—
3
p
2
x
+
s
u
5
u
fo r
atio n
itin g e q
h as fo r i ts
h
multiply
x,
ry
d
an
,
c
1
1
1
a
b
c
2
en c e
13
ab
r
q
—
2
‘
2
77
m
If
.
b,
a,
x
v lu
fi n d th e
a
Writin g
e o
th e
c are
3
f
r
v
d th e gi
3
‘
+b
is
en e xp res s io n
eq
ul
a
SI
we o b tain
is
If
u
q
e
v
x
th e
e
u
q
“
If t
h
P1
p,
x
c a se
e
o
"
al
e o
h
f S 3 in t is
eq
u
atio n
x
42
a t io n
is
.
u
n a lt e re
d by
c
h
an
i
n
g g
x
.
" l
+p
" 1
x
p
t wo
:
ast
re c i
e
u
q
0
7
r
p
s
n
_2 x
n
g
"'
2
a re
fl
u lt w h v
2
re s
i
..
u
al e
r
c
o
p
q
a
th e
flw
e
at io n s
.
p
x
fo r x ,
+
an
+ p ,,x
’
p
e
1
w iti
p
at o n
in t o
is
a t io n
b t ain e d b y
.
23 1
-
"
9
1
f mth l
l s s f
ro
o
u
es e
7
eq
v u
0,
‘
a t io n
p
u
en
to th e
(
c a lle d a recip ro ca l equ a ti o n
I f t h e gi
is
atio n
—2
S3
an
.
3
S 3 + 33 2
.
u
eq
e
—2
8
y + y
y
an d
56 8
c
3
v
2
Si =
en ce
Of
ts
3
‘
md
fo r x , th e tran s fo r
H ere
wh
ro o
—
3x
2
x
+
3
an
2p r
2
a
3
ge
2
.
E x a p le 2
an
0,
l
1
2
h
1
qy + p y
ro o ts
atio n
0
2
3
u
eq
1
3
by y
f th e
47 3
.
— r= 0
,
b
a
Write
qx
UA TIO N S
o
1
1
f
o
ts
ro o
OF EQ
sa
m
we
e,
n
v
+ p ,x +
1
:
mu t h v
s
a
f
ra c tio n s
0
.
e
1
P.
"
’
.
1,
p
d c le a rin g o f
P.
= =
h
n
2
p
_ ,x +
an
d t
R
h us
.
’
zu
we h a
v
’
e
47 4
If
i
1, t
p
h
at
( )
l
u
eq
n
GE BR A
—l
’
p2
—2 ’
pa
ts
z
p
n
.
p
z
ms qu idi t
f t er
o
al
ii
f mt h
t
s an
e
5
—a ’
ro
e
b egin n in g
.
1, t
—
pi
h
AL
en
p
z
If p ”
11
h
c o efli c ien
is , t h e
an d e n d a re
t
GH E R
HI
u
p
h
en
pg
—1 ’
n
m m s
m u s
m u
m
i
is o f 2
di e n s io n p m — p m, o r p m 0
t h e b e gin
I n t i c a s e th e c o e fli c ien t s Of t e r s e q idi t an t ro
a n d e n d a re e
a l in
a n i t de a n d o p o s it e in Si n , a n d
n in
g
p
g
q
g
iddle t e r
if t h e e q at io n is o f a n e ven degre e t h e
is wan t in g
if t h e
en c e
eq
hs
at o n
u
u
56 9
u
S ppo
.
s
e
th at
f( )
x
fi
f m
.
m
ro cal e
p
a re c i
0 is
:
u
q
.
a tio n
.
ss
an d o f an o dd de re e it h a s a
I f f ( x ) 0 is o f t h e rs t c la
g
1
If <
x
h
)
i
s
t
ro o t
1 ; s o t h at f ( x ) is di is ib le b y x
e
1( )
at io n O f t h e
s a re c i ro c a l e
rs t c la
i
o t ie n t , t h e n <
x
)
0
1( )
q
p
q
an d o f an e v e n de ree
g
z
v
u
.
fi
u
ss
.
I f f (x )
h as a
b e o re
f
e
an
v
1( )
x
( )
en
x
2
c la
degree
e
u
fi
s
=
ss
s
v
.
f
v
u
fis
.
an
e re o re
i
at o n s
57 0
du ced to
an
Let th e
ax
2
equ a ti o n
f
o
is
even
an
s
degree with
a
; w ic
y
Of re c ip ro c a l
m h hm
f m
s
.
recip ro ca l equ a tio n
A
.
rec ip ro ca l
y
o s i tive, o r c a n b e redu ced to thi s fo r
p
b e c o n ide re d a s t h e
or
t a n da rd
m
h f
v
v
.
ss
u
q
ss
an d o f a n e en
degree , it
0 is o f t h e e co n d c la
a ro o t
1 a n d a ro o t — 1 in t h is c a s e f ( x ) is di is ib le b y
1 , an d a s b e o re q5( x )
O is a recip ro c a l e q atio n o f t h e r t
a n d o f a n e ven de re e
g
H en c e
its la s t ter
t
s
an d o f an
0 is o f t h e e c o n d c la
O dd de re e , it
g
i
s di is ib le b
c as e f x
x
1
an
as
+ 1 ; in t i
d
,
y
( )
0 is a rec ipro c al e q at io n o f t h e rs t c la s a n d o f
hs
ro o t
I f f (x )
h as
=
equ a tio n
e
u
q
at io n
m+ b x2 m"
di idin g b y
x
’”
an
f
o
h a lf i ts di
th e
m
m
d fo r
s ta n da r
i
en s o n s
c an
.
be
+ cx
d
f
o
2
m
-
2
re a rra n
mw
i
n
t
h
e t er
g g
s,
e
ha
v
e
”
+
h
=
0
47 6
HI
m
E x a p le
u
eq
d
Fin
.
u d u
d u
h
ire
atio n
eq
;
c alc ul atio n
G E BR A
8 3x
in H
md
en c e
is p erfo r
o
21
7 6x
2
m
’
er s
as
e
.
o s e ro o ts e x cee
d
w
will b e o b tain e b y
atio n
eq
AL
wh
atio n
3 2 x3
4x +
re q
p ro p o s e
an d th e
eq
4
atio n
Th e
u
th e
GH E R
fo llo
s
ts
ro o
u
u
b s tit tin g x 2 fo r
we e plo y x + 2 a s
p ro ces s
o
f th e
.
m
s
32
83
76
4
24
35
4
16
s
m
4
s
13
[
b y 2 th e
0
4
4
d
x
in th e
i i s o r,
dv
21
[9
6
o
4
hu
T
md
th e tran s fo r
s
4x
Th e
ro o
th e p ro p o se
ts
o
eq
L e t th e
atio n
9
h
0,
=
u
e q atio n
is
or
h
if y
en
hi f
mv m
i
i
v
u
t
g
q
u se
e
c
o
e so
en
e
e
+
wh i h wh
c
,
(
n
Po l/
n
en
h
po
a rra n
+ 29
mt
t
h
h
at
Ob
—2
ta in t h e
ed
an
y
o
t
o
h
b t ain
+
n ew e
+ 29 2
h
1;
-
2
7
1
0
b
o
e re
mv
o
ed
u
q
+
?
(
is t h e
mt
if t h e t e r
o
be
x +
p
in de s c e n di n g p o
th e
en c e
ro o
ts
o
f
u
.
p
n
0;
at io n
”
70
7
w
f y , b ec o
e rs o
1 ) p ,h
n
s eco n
re
=
+
e
o
es
p
d, w e pu t
mv
m
np
h
o
p
0,
1
h
d is t h e t ird we pu t
0
k
0
so
.
be
2
d
0
us u
mf m
P. ( y
g
;
2
7
an
1)
f t h e s b tit tio n in t h e p re c e din g
a ss i n e d t e r
ro
an e
a t io n
g
q
09
I f th e t er
so
(x
2
+ 1,
o
on
a
h , we
x
z
9)
are
n
t
(4 x
2
atio n are
Th e
57 3
a rtic le is t o re
.
u
4
f t is
d u
eq
e
a
u
q
a
e r a s si n e d
g
(
n
_ 1
h
3
+
pl
22
dra t ic t o fi n d h ;
m
ter
.
)
an
2
mil
d si
O’
y w my mv
l
ar
e
a
re
o
e
TR A N S FOR M A TI O
mtim it w ill
f ll w i g mpl
So
E
xa
m
pl
L et
h
e ac
wl
il
be
o
8
a,
f th e
z ero
Re
e.
u
be
e
ex a
n
o
o
eq
mv
ts b y
al
to
.
2
s ec o n
ts ,
ro o
.
’
3p
47 7
ie n t t o p ro c ee d
en
d m m
fro
ter
h
t
so
at a
th e
md
h
in t h e
as
a tio n
h
T
en
th e
su
P
eq
e
u
g
u
atio n
c o efli c ien
is , th e
at
eq
+ fl+ 7
in th e tran s fo r
t
mti
u d
iv
u
57 4
o se
ire
re q
fo r x in th e g
tran s fo r
atio n
en e q
F o mth e
t
o
f th e
if we in c re as e
m f th
o
ts
e ro o
d mw
ill
ter
s ec o n
ro o
ts
e as
u
d
relat io n
will b
on
f( )
a t io n
x
with t
c o n n e ct ed
a re
n e
a
d b ub i u
fecte
e ef
t tin g x
st
s
y
.
eq
r
.
m s ig
so
v
co n
o re
U A TI ON S
.
H en ce th e
wh
th e
e
o
ro o
OF E Q
.
b e th e
7
,
m
be
es
e
N
:
m f m
0 we
a
o
r
y
i
e o f th e
n
e
g
hs
o
an
v
e
u
q
e
u
q
at io n
a t io n
by
.
u
h
u
L et y b e a ro o t o f t h e re q ire d e q atio n an d le t d
) x, y = 0
(
)
den o t e t h e a s sign e d relat io n ; t en t h e t ran s o r e d e q at io n c an
b e Ob t a in e d e it e r b y e x p re s s in g x a s a
n c t io n
an s
of y b
e
y
o f th e e
a t io n 4) ( x ,
0
a n d s b s t it t in
h
i
s v al e o f x in
t
y)
g
q
= 0
w
x
b
li
i
n
a
i
n
b
h
n
0
o
r
e
t
x
e
t
e
e
n
t
e
e
a
t
i
x
o
;
f( )
f( )
y
g
q
an d )
x,
4 ( y) 0
h
u
m
:
u
:
u
fu
f m
u
m
u
s
u
.
m
E x a p le 1
If
fo r th e eq atio n
m
u
wh
.
b,
a,
th e
c are
o s e ro o
ts
ro o
ts o
f th e
u
eq
are
atio n
x3
1
be
W
h
h
dt
an
=
en x
v
in th e gi
a
erefo re
en eq
u
atio n
E
be
ab c
u
eq
e
=
y x+
hu
t
s
th e
u d u
req
m
ire
eq
atio n
mth
E x a p le 2
Fo r
ifferen ce s o f th e ro o ts
d
.
o
u
eq
a,
atio n
b,
are
c
b e th e
2
u
u
e eq
f th e
ro o
b
(
w
atio n
’
01
ill b e
md
th e tran s fo r
o b tain e
d
e
eq
u
atio n
b y th e
s
ub u
th e
u
are s
s tit
;
tio n
x
is
c
ts
o
)
c
wh
atio n
b ic
3
x
L et
in
a
1
md
th e tran sfo r
=
y
,
x
q
f th e
2
,
(
c
r:
0
u
b ic
c
a
)
ro o
o se
ts
are
sq
o
f th e
.
2
,
h
t
(a
th e
en
b)
2
.
ro o
ts
of
th e
ui d
req
re
47 8
HI
b
(
c
)
2
GH E R
—c 2
= b2
G EB R A
AL
2
2
=
2b e a
b +c
.
2
2
a
2 q — a2 l
-
w
h en
als o
eq
u
atio n
x
=
in
a
v
gi
th e
;
u
eq
en
—2 —
q
y
hu
T
hv
we
s
a e
mi
eli
to
3
x
x
By
s
u
u
u
S b s tit tin g an d
re
y
C OR
If a , b , c are
2 7 r2
4 q3 is n e gati e
v
.
H
y)
3r
x
g, we
6 qy
2
c
)
2
,
(
If
t
u
iv
h
th e
at
u
2 r= 0
c
a
) (
2
,
e,
3
x
in t egral
or
ts ,
t
.
v
So l
e
th e
e
u
e
eq
al ro o
u
ts
.
.
4 —
x
3
1 0x
5
.
x5—
4
5x
6
.
4 x6 —
3
4
d
u
n
atio n
h as
a e
a
all
ere fo re
its
v
h
v
3
e n egati e
s
o n e ro o
e
e
on
a
a e
s
pro
c an
du
XXXV d
.
2
3
x
4x +
at o n
3x
er
s
rs t
4
on s
+ 2 6x
2
9 x3 —
'
5
t
t
z ero
h as a n egati
t wo i agin ary
m
ro o
ts
.
erefo re
t
,
.
ity fo r t h e
e
e
o se
+
v h
p o s iti e
my h v
m
u
t b
g
+ qx + r = 0
u
wh h
4
2x + x3
are all
md qu ti
mu t h v
e e qu atio n
an
t
a
b)
a
2
(2) ( )
is
at
f m h qu i
wh fi t mi u
t
i
u
q
Tran s o r
c o effi c ien t o f
2
th e
fi
.
e ro o t
ro o
ts ,
c e a n egati e ro o t
in
v
.
f mth
c o ef c ien
0
2
.
.
atio n s
4 q3
27 r
E XA M PL E S
1
e
.
2
atio n
eq
v
h
Tran s
md
th e tran s fo r
tain
md
uh
md u
.
eq
x
3 is
4
+ q
po s iti e , th e tran s fo r
A
r
e refo re th e o rigin al e q atio n
t
t
[
a p air o f ro o ts
ic
Sin c e it is o n ly s c
th e tran s fo r e eq atio n
2
27 r
u
th e
0,
9g y
3=
4
0 th e tran s fo r
+ q
o rigin al eq atio n h as two e q
If
ob
in
+
r
z
real, ( b
e n egat
st
2
27r
y)
c in
3
qx
,
.
d h
mu b
o r er
2 7 r2 + 4 q3
real
th e
in
en c e
du
fv
een
+ (2 g
(q
b trac tio n
2
w
b et
n ate x
3
atio n
2
—
=
y
(b c)
24 x +
2
9 x + 5x
4
57 x —
73
3
x
2
5
+ 7x
n
lx
0 in t o
zl
c o effi c ien
5x3
ity
.
.
+
x
2
t
o
x
an o
t
f th e
fi
+1
0 in to
=
rs t
h wi h
t
er
m
t er
.
an o
t
hr
e
480
HI
GH E R
AL
GE BR A
C UB I C E Q U A TIO
y
57 5
Th e ge n e ra l t p e
.
t
u
as ex
mpl
er
To
x =
so
x
ak e a s
d t h e gi
v
v
l
en
2
‘
y
eq
u
h
3
h
s
en
e
+ z
a t io n
z
t y,
x
q
u
q
s t an
r
R
S
.
u
q
a t io n
is
0,
=
u ti
b
du
0
d f mf
ub i q
on
a
c an
e re
ce
,
da r
o r
3
a t io n
x
3y z ( y
+ z
o
x
q
a c
u a t io n
c e
.
0
r
.
3
3
+
b ec o
+ e
a re
s
)
m
es
+
qu
w
t
o
y
e
al t o
q
an
m u
h
fu h
s
m y
hy
an
s
f th
th y
tit ie
s
ub j e
to th e c o n
o f th e
i
n
e
g
ct
is
on e o
e ro o t s
dit io n t a t t e ir su
w
e
a t io n ;
if
e
rt e r s u p o e t h a t
s a t is
t
h
e
e
p
q
a re c o
r
i
n ate
l
e
l
t
d
e
t
e
e
3y z
0
t
e
t
s
W
e
,
q
p
u
d to th e
en
y
A t p re
3
th e
th e
e
; t
+ z
y
h
.
a
x
an
c e
eq
.
L et
ub i
l
a in e d in A rt 57 3 t is
p
e
57 6
—
l
n
+
or
c
a c
Q
f m
wh i h w sh ll t
si
f
x +
x
b
3
o
N
.
m
fy
.
e
hu
u
q
ob
v
a t io n
t ain
3
7
h
en c e
y
s
,
z
3
a re
th e
ro o
ts
"
yz
u
f th e q
o
9
27
3
3
a
dra t ic
3
q
27
So lvin g
e
u
q
i
at o n
n
a
,
u
d p ttin g
f
2
3
’
7
we
o
b ta in t h e
v
v lu
a
e o
u
f
x
9
a?
f mth
ro
e re la t io n
x
y
z
t
;
y
hu
s
Th e ab o e s o l tio n is gen e rall kn o wn a s Ca rda n s So lu tio n ,
a s it w a s
rs t
i
b
l
i
i
n t h e A rs M a n a , in 1 54 5 C a rda n
h
s
e
d
b
p
y
g
Ob t a in e d t h e
o l t io n
Ta rt aglia ; b u t t h e so l t io n o f t h e
ro
c b ic s e e
s to h a e b een
du e o rigin all t o Sc ip io
e rre o , a b o t
fi
u
m
u
s u
v
h
m
f m
’
y
u
F
.
u
C
1 505
o n d
.
fu
UB IC
E Q p A TI ON s
48 1
.
h
h
uj
w
in t e re s t in g is t o ric a l n o t e o n t is s b e c t
ill b e
’
a t t h e e n d o f B rn s ide an d Pa n t o n s Th eo r
o
E
u
a
t
i
n
o
s
y f q
A
n
u
.
h
u
h
h h
h
v u h
ub
By A rt 1 1 0, e a c o f t h e q an t it ie s o n t h e rig t an d
57 7
2
an d
o f th e
re c e din
a rt ic le h a s t ree
s ide o f e q a t io n s ( 1
p
g
( )
)
e n c e it
o ld a pp e a r t a t x h a s n i n e
al es
c b e ro o t s ,
t
i
s
,
;
.
u
u
hwv
o
e
.
h
is
e r,
wu
n o
t th e
c a se
s
-
Fo r s in c e y z
.
h
u
g
,
th e
c
e ro o
ts
a re
f e a c h pa ir is ra t io n a l
a ir Of c b e ro o t s
a l e s o f an
H en
i
h
c
p
y
l l t is c o n dit io n , t h e o n l
will b e
o t h e r a d is s ib le p a ir
g
2
g
( a e an d w
a re t h e i
a in a r
m
w
z,
e re w, ( 0
c b e ro o ts o f
,
,
y
y
g
n it
H e n c e t h e ro o t s o f t h e e q a tio n a re
to b
t a ke n in pa ir s o t
de n o t e t h e
c e if y ,
e
z
fu fi h
u y
t h e p ro d
—
at
v
u
u
.
z
:/
E
xa
m
pl
e
v
So l e
.
h
Pu t y + z fo r x t
,
th e
,
u
z3
3yz
15: 0,
3
y
en c e
z3
y +
als o
h
31
,
2
are
th e
ro o ts o
f th e
t
y
3z 3 =
eq
3=
1
2
wy + w z
-
ro o
ts
s
.
.
=
12 6 ;
u
1 25
atio n
1 2 5,
2
3
z
J
-
—3 + 2
are
x
= 1 26
=
y 5,
d th e
]
2
(o z
u
wh
1 2 6 t + 1 2 5= 0 ;
2
an
£0
1 5x : 1 2 6
1 5)
en
3
2,
+ ( 3y z
pu t
3
x
3
2
y
2
J
=1
.
—3
J
TS
,
—3 ,
—3
why
-
2
J
,
-
3
.
y
w e app a re n tl o b t a in
To e x pla in t h e re a so n
57 8
a re t o b e
v a l e s fo r x in A rt 57 6 , w e o b se rv e t a t y a n d
.
u
.
f mth
ro
so
e eq
u
u
l t io n t h e
H H
.
.
A
at io n s
s ec o n
.
.
en
y +
h
(0
eq atio n
3
t
y
(b
2
u
m
m
y
wh
l
ct o
d
y
o
3
ft
z
h
3
ese
r:
h
z
c
h
3
an
ed
g
in t o y
z
in e
n d
fu
s f
wh i h
o
b u t in t h e p ro c e s
O, y z
wa s
n
o
c
482
HI
w u ld
lu
o
es o
va
fx
u
a
p
z
d
ro c ee d
re
p
re se n
fi
o
t
h
rs t
f t
o
two b
er
h
t t
h
ec o
m
( )
th e
ro o
+
4
ts b
ec o
p
re s s io n
ts
ro o
th e
c
s
9
2
7
—z
y
is
m 2y
e
u
+
4
o f th e
h
en
y
3
g
h
er
ro o
ts
o
f
ub
z
h
b ot
a re
ro o
e
w y + wz
su b
d by
3
t s, t
h
re al ;
th e
en
le t
ro o
ts
2
wz,
+
d
an
c al c
e
.
u ti
s tit
z e ro
t
,
9
is
27
u
n e
r w
f
o
g
n
g
+
in
en
y
+
v h
ib
a +
or
e,
t
an
d
3
3
= z
m— i
i
m
(
n
y
or
n
)
in
—
w
)
or
,
o r
a,
an
t
an
2
d w th e
s
j
.
c ase
y
= z
2y , — y , — y
or
d
n
hi
3
z
.
y
n ar
a
d
an
,
mgi
i
a re
u
.
d
an
2
;
3
— ib
in
in
w ),
,
—
y
e
en
a
m
a re
+ z
2
w
y(
ati
f m
m
m
h
w
y(
,
y
_
J 3
3
f t es e q an t it ie s
b ic b e c o e
o
th e
o re
h mti
wy
3
i
i
i
I
f
( )
h
t
e,
re a l, a n
+
2
If
t
e
g+ z
ii
v
e ir a rit
+ z,
is
e se
o
.
y
Th e
m fu lly
c o n s ide r
to
is p o sit i
a re
th e
en ce
u
3
27
h
"
0
r
9
—
If
i
()
u
x
.
Q
o r yz
i
3
o f t h e c b ic s
_
l t io n s
a re s o
G
A L EBR A
(”
ld if y e
mq
.
y
o
We
57 9
at io n
e
q
an
h
a ls o
GH E R
h
ex
u
S pp o s e t at t h e c b e
in ; t en t h e ro o t s o f
m
h
m;
—m
—m
+
2
-
n
S
J ;
n
3
J ;
ll
l qu
wh i h
t it i
i
l
A
h w v
th
g
lg b
it h m i l
i l mt h d f fi di g t h
l
f
t
th
ub
t f imgi
y qu ti i s [C mp A t
th
lu t i
b t i d i A t 57 6 i
wh th
f lit l p
ti l
ts f th
ub i
ll
u
l
d
l
q
i
mtims ll d t h I d ib l C
Th i
s
fC d
a re
c
e
c
e
so
on
ro o
o
O
or a
ro o
e c
uti
on
58 0
co
s
.
n ar
a
n
r
o
e
o
o
an
t
s
.
c a re a
s so
e
ra c a
e
s
e re
er
re a
e
ca
n
t
an
e
u n e
e
e
n
e
o
n o
e n e ra
ra c
a
rre
r
a re
o
e
ex ac
e
.
u se
ca
va u e o
e
en
.
uc
e
ar
as e o
an
’
.
j u st m t i d t h
m y f ll w L t th
I n t h e i rredu cib le ca s e
l
t
d
b
i
e
e
T
r
r
n
o
o
e
t
a
s
p
y
g
m
.
be
e
o
a n e
s ca se
so l
es
an
'
et ca
ar
re a
a
(
a
ib )
é
en
o
(
o
a
s
ib )
on e
e
e
u
u
m
l t io n
a
y
s o l t io n b e
I
e so
484
mi
n a t in
e li
by
GH E R
HI
g
f mt h
db
a an
ro
2 h — qh
F mt h i
fu
o
n
c
s
ro
d [A rt
ub i
t
.
2
c
-
e
v lu
d th e
a
f
es o
xa
l
m
p
e
v
So l
.
A dd
a x
2
x
h
t
en
u
by
eq
b
+ 2a b x
4
— 2 x3 +
atin
(
a
2
2
(
+
eq
to
e ac
)x
a
n
0
=
r
.
f 10
A ls o
e o
.
w ys b
al
c an
a
e
b) ;
2
dx +
f mth
b t a in e d
2
ro
)
e
t wo q
d
ass
u
a
dra t ic s
0,
:
atio n
2x
—5
o
(
:
—
qs
v lu
w
—b
2
— 5x 2 + 1 0x
3
h d
si e o
f th e
eq
u
atio n
,
an
um
e
2
—
=
3
+ 2 ( a b + 5) x + b
(x
2
hv
g c o effi cien ts , we
2=
a
h)
2
ta in
2
kn o
a re
Ob
7
0
9) (
2
p
4
x
2
to b e
u
th e
e
2
db
we
a t io n s ,
o n e re a l
+px +
x a re
x
E
2
.
—
a t io n
s a an
x
+ 19
u
2 (p r
1
u
q
hu
(
an
ese e
q
2
7
4
(
7
(2
9 0
3
G
A L E BR A
2h + 6 ,
a e
k — 5,
ab
b 2 = k2 + 3 ;
( 2k + 6 )
2 k3
h
um
By trial, we fi n d t
B u t fro
mth
e as s
at
x
al es o f
x
h
wh
at
is ,
en ce
x
th e
58 3
.
u
S pp o se t
s um
a s
e
x
4
h
at
+
2
a an
—x
s
3=
n
g
so
i
: ;
d
+ rx + s
4, b = 4,
ab
b)
ax
2
hv
a e
u
th e two
eq
atio n s
( 2x
x
2
+x
J5
2
u
l t io n wa s gi
t h e b iqu a dra t ic
2
(
2
at
d b , we
2
o
x
q
2=
1=
-
t
s
k)
x
k,
w h
an
f ll w i
o
en c e a =
2
ro o ts are
Th e
2
v u
u
S b s tit tin g th e
t
2
p tio n , it fo llo
(
u
h
1
k
— 4k
5
k
+
2
(
x
2
e
u
q
v
en
a t io n
b y De
is
u
re d
s
c a rt e s
c ed
in 1 6 3 7
to th e
.
f m
or
U A TI O N S
B I Q U A D R A TI O E Q
t
h
by
en
u
q
e
a t in
F mt h
ro
g
m
— hg z
l+
fi
e
h
en c e s
ub
s t it
mh
=
ui
t
q
n
h
es e e
+
q
+
h
h
u
"
u
q
’
2qh
u
q
e
a t io n s ,
)
a t io n
s
.
we
b ta in
o
7
(q
wh h
hu wh
u
u
7C
,
r
43 ) h
2
-
‘
)
l
c
q
r
“
z
2l= h +
h
h
q
m
l
2
i
n t h e t ird
g
7
0
(
e
= r
,
r
2
3
h
f t
o
hv
a
—l
m
(
)
h
,
t wo
rs t
2
t s , we
fi c ie n
c o ef
485
.
2
y
0
r
.
v
u
ic alwa s h a s o n e re a l p o s iti e s o l
T is is a c b ic in h
2
s
is k n o n t h e a l es o f l a n d
t
e n is
t io n [A rt
in e d, a n d t h e s o l t io n o f t h e b iq a dra t ic is Ob t a in e d
a re de t e r
b y s o l in g t h e t wo q a dra t ic s
2
m
.
v
an
m
v
So l
E xa p le
.
th e
e
eq
u
A ss
h
t
en
um
u
e
by
eq
atin
-
m— k
2
we o b tain
u
a e
k
8) ( h
m
l
)
(
3
2k
m=
l
8,
z
3;
1 2 702 ,
8)
4
2
—
4k l 16k
k
h
u
--
d wh
v u
h
T is e q atio n is c learly s atisfi e
ffi c ien t to co n s i er o n e o f th e al
d
m+ l= 2 m l= 4 ;
,
hu
T
s
x
4
en ce
an
hv
6
-
h
2x + 8x
2,
2k
or
s
g
2
4
g co e ffi c ien ts , we
-
en ce
u
m
— 2x 2 + 8 x
l l
wh
x
v u
atio n
x
4
x
d
w
h
I
erefo re
584
th e
t
2
at
o r
u
we
p ttin g
is ,
k:
m= 3
l = —1 ,
hv
a e
It
w
ill b e
.
— 1 x2
)(
+ 2x
an
J2
1i
ro o ts are
k
en
fk
— 2x 2 + 8 x
x
dt
es o
2
d
x
2
J
,
2
.
u
u
Th e ge n e ral algeb ra ic a l s o l tio n o f e q at io n s o f a
’
degree h ig e r t an t h e o rt h a s n o t b ee n Ob ta in e d, an d A b el s
de o n s t ra t io n o f t h e i p o s s ib ilit o f s c a s o l tio n is ge n e ra ll
f
an
i
n
t
s
o
a c c ep t e d b
M
at e
a t ic ia n s
h
c
e
f
c
r
t
e
o
e
I
f
o
e
e
,
,
y
nd
o
a t io n a re n
a
e
b
e
e ric a l, t h e
l
n
rea l ro o t
a
o
f
a
e
y
q
y
’
t o a n y re q ire d de gre e o f a c c ra c
b y H o rn e r s M e t o d o f a p
n t re a t ise s o n
ro x i
a t io n , a
d
i
ll
n
a
c
c
n
w
il
l
o
e
o
f
w
i
b
t
o
c
p
t h e Th eo ry of E qu atio n s
m
u
.
h
h
fu h
hm
um
u
m
fu
m
y
u
hwv
v u
u y
h h
h
u
u
.
fi
m
.
f
u
y
h
fu
HIGHER G
lu d w it h t h
4 86
AL
We
58 5
.
n eo n s e
E
xa
u
q
s
h
a t io n s
m
pl
e
1
a ll c o n c
v
th e
e
eq
u
y z
a x
2
b y
2
e z
d
3
3
3
c z
d3a
b y
A ss
p
t at t
e
u
,
2
e q atio n s , b e gin n in
'
r b ein g q an titie s
ese
s p ec tively
q
ey
,
are s
u
uh h
ils t b ,
d
c,
th e
are
h
hu
dt
T
ts
ro o
by
sy
th e
e try
mm
Co n
f th e
o
(a
b)
v u
al e x
is fo
ud
n
mth
g fro
eq
eq
u
u
(
dw
o
n
by
sy
b y
3
a x
b 3y
vm u
m
al
—b
a
e o
f
)(
t
th e
—c
)(
a
d)
x
)(
a
—d
m
E x a p le 2
(
x
.
.
v u
al es o
fy,
z
,
be
u c an
w
dw
ritten
e try .
i’
d
o
n
3
c z
W
W
u
,
3
da
,
—c
)(
—b
d
ud
v u
n
an
,
e eq
v u
d th e
atio n s
—c k — d
k
)(
)
)(
al es o
f y,
z
,
.
u
h as b een facilitate
d
c an
be
S
.
a
)(
hw h
x
t
e
th e
at
(
—e
u
b)
x
) f (
2
-
ts
ro o
f th e
o
—a —
)
x
2
g
eq
(
u
atio n
b ) — h 2 (x
x
e
v
gi
—a
en
eq
atio n
tv
b)
a
v— 0
ll
’
L e t p , g b e th e
ro o
(
ts
o
)
we
,
~
a e
f }
f th e q
(
hv
2
a
c—
ud
a
b) ( x
{9 (
2
b l+ h
x
ratic
c
—
) f
2
=
o,
2
(
e
-
C
l
w
b y th e
.
mth
en
d
7c
are all re al
Fro
n e
);
.
—
mi
e ter
n
a e
is fo
x
u
m
(
ud
v h h
r re
k,
hv
a
Th e s o l tio n o f th e ab o
U n deter i n ed M u ltip liers
x
b y 1 , p , q,
+ by
a x
d
th e
) (a
an d
,
e st,
atio n
g
b y p ro cee in g as b efo re , we
s
c
.
w
atio n s are
fi
hu
(a
lo
e
.
ax
T
s ce lla
e
.
:
,
If th e
.
m mi
k
—b
ere fo re
s
f so
ic
are at p re s e n t
co effic ien ts o f y , z , u
an is
H en c e
an
o
O,
wh h
th e
at
t
c
a =
x
wh
s s io n
O,
u
2
a x
h
um h h
u
by
ax
u
dis c
e
atio n s
x
M ltip ly t
.
.
So l
.
e
E BR A
2fgh } = 0
ritte n
u se o
f
HIGHER
h u wh
d
d
v u
mu
u
h
d
v u
mu
d
4 88
AL
GE BR A
.
d
v u
en c leare
o f frac tio n s i s o f th e s ec o n d
T is eq atio n
irt
i s s atis fi e b y th e th ree al e s
dz u , d = v, in
en ce it
e q atio n s ;
s t b e an i en tity
[A rt
th e
To fi n
al e o
fx,
ltiply
b
a
{I}
h
t
at
:
By
sy
mmt y
e r
we
,
hv
a
(
a
(a
b)
a
(
1
.
3
.
e
x
c
-
c
—a
1 8x =
Pro
.
)
v
9
e
.
11
12
13
t
e
th e fo llo
a
+
)
)
a
c—
b)
)(
)
XXXV
.
e
.
.
atio n s
2
35
.
4
.
x3
1 7 20 = O
.
.
3
x —
.
l 5x
2
+
3
2x —
.
.
4
x —
3x
2
16
.
4
3
r=
u
4
d
x
1 2x
12
O
7x
17
v
3
x
14
4 —
x
2 x3
2 0x — I G= O
.
4
.
O
.
—
1
x
x
1= O
+ 7
+6
2
.
192
w
relat io n
my b
en c e s o l e
2—
l ox
2
8 0x
th e
0
10
4 —
d
.
2 0x
9x
n
eq
3
atio n s
.
6x —
+g +
e
— 6x — 2 = O
é —
z
x
n
4
Fi
f th e
atio n
— 8x
x
9
+
+
4
x
.
ro o t
o
2
3
8x
.
x
th e
at
.
x
.
re al
— 3 x 2 — 4 2x — 4 O= 0
4
x
4x
H
pu t
u
wi g qu
.
18
—V
en
.
15
17
eq
—
b
(
6
v h
f/4
So l
3
n
a
a
h
.
is 2
x
o
2 8x3
.
8
3
o
c
(
dt
an
)
4
5
7
f ll wi g
th e
+ 0,
u)
) (
0
E XA M PLE S
v
v
e
b
(
So l
a
a
b
(
is ,
by
l(
e
.
.
up
in 0, an d
o f th e gi en
egre e
h hh
O, w ic
b e t een 9 an d r in
4
t
o th e fo r
u
t
i
n
x
p
a
e
th e
eq
u
atio n
8 x3
m
:
ul
d h
as e
q
o r er
a
ro o
ts
t
at
th e
.
eq
u
atio n
B I Q U A D R A TI O
19
hv
a
If
.
e a co
x
mm
3
3 gx + r
fac to r,
on
(p
4
If t
hyh v
t wo
e
co
hw h
t
e
2
mm
g—
f
t
o
e
.
as a
rat c
a
t
e
o
.
e
o s e ro o
23
e e
e e
at o n
ts
2
s
6
n
.
t
o
t
e
t
o O
4
1
2
th e
er
t
t
e t er
s
h
e
9
2
s
on
t
h
a
ct o
e
r
3
.
5
x
d
o
=
r
p
bc —
(
ac
+p
3
x
2
eq
ad
b
?
ul
ro o
a
ts ,
)
gx
z
rx
s
z my b
0
a
e so
vd
l
e
2
3 2x + 8
=
O,
ts
are
f th e
eq
u
atio n
1
B
,
2
m
v h
+ s = 0, p ro e t at if t h e s u
3
o
t
e
u
o
h
e
o
t
e
t
w
o
9
4
8
r= 0 ;
t
s
ft
r
+
1
pq
t o o f th e ro o t s is e q al t o t h e p ro
ct o f
3
x
+ gx
m
rx
h
u
du
2 9 x + 56
=
O h as t wo
ro o
ts
wh
o se
p ro
du
ct
is
are
5
x
s o
th e
ro o
4 09 x + 2 8 5= O
ts
wh
o se su
mi
s
no
w
5
.
of
+ p n _ 1x
at
Th e
e x plain
o
—
at
.
su
m f tw
o
o ro o ts o
x
is 4
le ge
t
e
+ 2 8x
“
28
9
2
hw h
.
o ro o
e
I f a, b ,
.
wt
s e
a
t
( pg
.
4—
x
on
ro
ne
n
.
27
ts
e
2
o r
e e
:
a
4
x
ro o ts o
o s e ro o
e e
e ro o
q
.
1 8x
a re
on
t
.
n
t
at
t
ot
an
a
eq
)
to
a
u ti wh
h
I
24
q u ti
p
i
u
l
h
f w f h
q
f w
if h p du
d h
h
h w 1
i
u
t
0
25 Th
q
u i y d mi h m
F i d h tw
t
f
26
fi n d th e
2p x
h a s t wo
at o n
s
I f a , B, y , 8
.
1s e
at t
2
r =
x
one o
x
at o n
eq
a t e ac
t
e
p
0
F .
u i
h w h h f h m qu l
Sh w th
h
21
qu i
u
d
i
if
q
p
h
S lv
22
qu i
+ 6
i
f wh
6
.
d
r
ac to rs , s
on
0 —
1
an
?
489
.
at
q) ( a
2
I f th e
20
s
a
e
s
U A TI O N S
EQ
h f
f t is
wh y
ac t
4
o n at t e
th e
8x
3
g to
ails
o
n
.
u
lv
eq
2
2
1
x
+
mth d f
e
P2 + P4
f th e
mp ti
—
P3 + P5
at io n
20x + 5= 0
so
"
e
th e
eq
u
atio n
fro
mth
e
k
M I SCE LL A N E OU S E XA M PLE S
.
t m
i
l
um f
v
f
p
y
hw h
i hm
(
m d p du
u mb u h h h i diff
Fi d tw
2
h
7
i
d u b l d b y v i g th digi
l
f
t ti
I wh t
3
i
h
u
S lv
4
q
1
ar
I f 3 1 , 3 2 , 3 3 are th e
et ical p ro gres s io n , s
.
t
are
to
o n e an o
n
.
t
x
+ 2)
on
(
+ 3)
x
4)
x
y(
In
.
A P,
m
an
fi rs t p t er
h
.
.
O,
s =
3
at
t
v
So l
e
th e
s
f
ew t
o
eq
u
( 1 ) (a + b ) (d x
( 2)
(
s
uh h
.
s e rI e s
8
e
whi h
h h
c
at
t
e
Fi
n
-
l b)
(
th e
a
9
.
10
.
Fin
:
d
a
th e
ro
c t,
ts ?
e
a
‘
su
m
m f th
o
e
( r+ q>
9
bx )
a
1
M
R
A
[
(
2
1
2
6
2
‘
a x
) (a + bx )
.
WOOLW I CH ]
.
.
.
h
a
an
ts
o
?
res s o n
r
fx +
2
B+ B ,
d Qy
e o
ND C V S V ]
i
w
t
h
fi
mi u i y
g
h i t y f u th t m f m g m i
fi
d
h
l
t
v
u
f
0
g
p
I
[
{1 2 ( x
ro o
v lu
a
m
1
hm
d
I f 2x
an
atio n s
5
3 :
( 2x
are
e rs n
2
x
(
fi
m
l
If a , B
o
an
i
is t h e
rs t t e r
t
he
f
,
o f t h e n e xt
t
r s
su
e
9
a
.
.
re
e
o
z
d an arit
e tic al p ro
t at th e s e co n , t en t an d t
c
7
ere n c e , s u
.
l
5
x
e
x
p
.
ec t
s res
2
e r
25
s
a
6
er
.
no a
2
( )
5
2 a , 3 72
,
at o n s
e e
(
n
at
t
c
24
,
sca e o
e
(
1
( )
1,
er as
a
o
.
t
t
e
e rs s
o n
n
.
o
s
s
0
b
3
o
-
o se
+3
b
"
0
,
1
,
4
+a
s
n
,
2
Z
IL
ter
ER
s
or
a
a
es o
e
eo
I CE
n
t
e tr c
4
B +B
fin d th e
I
rs
er
r
x
3
IA
v lu
a
.
e o
f
f
5
V1 5
V3 13)
;
A
M
R
[
.
a an
d
mgi
th e i
a
4
a
+3
n ar
4
c
y
a
‘
l
ub
fi
‘
l
e ro o ts o
0
.
f
u
.
n
ity ,
.
s
W O OL W I C H ]
.
hwh
e
t
at
492
HI
d
Fi
m f th
GH ER
G
A L E BR A
u
.
k
d c ts o f th e in tegers 1 , 2 , 3 ,
77 t a
alf t h e e x c es s o f th e s u
t wo at a t i e, an d s e w t at it is e q al t o
o f t eir s q u ares
t h e c b es o f th e gi en in t egers o ver th e s u
23
n
u
7
a
su
v
o
h
m
h
o
r
p
e
u
h
m h
um
w k
fm
mf
.
en
o
.
v
d
d
w w
w ud
f
w k
d
w d
e 20 lo a e s o f b re a
d h is a ily c o n s
in a
ee
5 p e r c e n t , an d th e p ric e o f b rea we re rais e
ere rais e
ages
B
t
i
i
s
i
n
a
ee
u
f
h
a
es
e
e
l
o
a
d
oul
r
re
r c en t , h e
6
e
e
g
g
p
f
ell 1 0 p er c en t , t e n h e
o l
l
o se l d
r c e n t , an d b rea
e
p
f
q
ee
: fi n d hi s wee ly wages an d th e p ric e o f a lo a
24
I fh is
2
th e
m
.
A
w
.
an an
w
3w k
25
.
26
Th e
du
f th e
b ers
ct o
um
S lv th
o
a
an
e
e eq
(b
2
)
e
I f Va
.
.
.
.
is t o
es
-
in arit
t h e p ro
hm i l p g
du t f h m
et c a
c
t
o
res s io n
ro
e
is 4 8 an d
2 7 to 3 5
ean s as
.
W
2
( )
m
e x tre
h
.
ers
our n
o
su
n
.
.
.
.
1
( )
27
w d
d
k
mff
u mb
.
t h e p ro
fin d th e
d
.
x
u
atio n s
+b
(
c — a
)
x
b) = 0
+ c (a
.
M
A
T
IL TR I P OS
[
]
.
Vb
—x +
d if
—x=
s
h w th
e
0,
S
hw h
t
e
at
at
m
hu
hu
h h
m
v hu
h
u h
m
h
ju
i
th
u
S l
29
q
h
d wh h
m
h
d
A train , an o r after s tartin g,
ee ts wit
an acc i e n t
ic
det ain s it an o r, aft er w ic it p ro cee s at t re e fifth s o f its fo r e r
o rs aft er t i
e : b u t h a d th e a cc i e n t
rat e an d arri e s 3
ap e n e
5
0
p
iles fart e r o n th e lin e, it wo ld ave arri ed 1 5h rs s o o n e r : fi n d th e
len gt o f t h e o rn ey
28
d
.
h
-
v
d
.
.
o ve
.
at o n s
e e
7 x = 6y ,
2 x + y = 2z , 9 z
M
R
A
[
.
m
.
.
WOOLWI CH ]
.
h m mh mi
v
v d ly h
at
30 Six p ap ers are s e t in ex a in atio n , two o f t e in
in h o w an y iffe ren t o r ers c an th e p ap ers b e gi e n , p ro ide
atic al p ap ers are n o t s c c e s s i e ?
th e t wo
at e
.
31
.
d
m d
mh m
I h wm y w y
c o n s is t in
32
my h
.
a
an
o
n
f
g
o
Fi
n
h
d
ave a c o
alf c ro
-
w
a
an
s
n s, s
d b
mm f
a
u
so
can
h
th
at
.
In
do it in s ix
th e ti e ?
.
x 3 + x x3
u
m
o
rs
e
o re ,
o
a
t
at
-
+ 1 1 x + 6 an d
2
x + x +
p
g
m
f th e fo r
wh tim w uld A
hu m B l
at
on
d
.
.
h d
hu m
B , 0 to get
o n e in o n e
o
,
x
3
2
+ b x + 14x + 8
.
ND N UN
w k if A l
L
O
[
33
at c s
£5 4 3 2 01 b e p ai in e xac tly 6 0 c o in s ,
illin gs an d fo rp en n y p ieces ?
ac t o r o
on
v
e
O
er
o a
r
o re, an
or
d0
Y]
IV E R SI T
.
uld
wi
a o n e co
alo n e
in t
ce
M I SC E L L A
34
p
ro
If tl e
.
v h
at
35
Fin
t
e
.
s io n
o
9
g
eq
b}
+
d by
1,
an
2 x + 2x
2
v
So l
)
th e
e
by
d
Fi
n
.
x =
y
mi l Th
a
ro o
d mit
39
S
v lu
th e
d
e o
a
u c tio n
I f a, b ,
.
h w th
e
f re
hv
1
a
ly
o n e so
u
l tio n
=
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mth
e
fi
mi
fi ve t e r
rs t
2
f x +p x + g= 0 is th e
o
.
s
n
th e
ex
p
an
sq
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f th e
a re o
o
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h
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CA M B
.
COL L C x
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C O LL
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N
S
.
-
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ce
it to its lo
z are real
c,
d
an
q
u
an
w
m
te r
es t
tities ,
7
an
0, y = 0,
[M A TIL
.
TR I
PO S ]
th e
u
val e o f x
mi
is th e greates t t er
is
6
y
2
—z2
)
,
0
z =
.
I
H
R
ST S C OLL C A M B
C
[
n
th e
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7
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C OL L CA M B
.
.
t
c
at
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fa, b,
c
.
[SI
AM
.
2
+
y
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DN E Y COLL
y
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.
CA M B ]
.
.
[COR
If x , y ,
z are i
n
lo g
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PUS
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30
( 1)
( 2)
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e su
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OLL
1+b +
si
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f 1
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u h h h i mmul ipli d b y h m
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ml ipli d b y h diff
s C C B]
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b e rs s
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41
s q ares is 5500, an d t
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40
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t h e frac tio n
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P
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M
R
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x
a
E x A M PLE S
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=
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38
US
2
.
.
d
th e Bin o
l
If o n e o f th e
36
3—
s h o w th at p
g ( 3p
37
EO
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a
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at io n s a x
N
at
COL L
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4 94
HI
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45
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GH E R
AL
GE B R A
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at
1
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MM
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46
t
h
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3a — 2 b
If
.
will
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3e + 2x
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3b
3a)
5
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H R I ST S C O L L C A M B
C
[
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h
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t
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ifferen t ) at eac en d ?
c o n s o n an
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e
.
as
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vw
d fu
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vw
mdd
b
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d
d
h
( p
t
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w l
d
v
d
wh i h 6
h
48
;
p
i d b y wi
h vi g v d g i h m q u i i w
h f m
m y i w b f l b y d h w mj i y w
C
i d ?
J
N
B
h wm y h g d h i m
S
C
8
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Sh w h t
49
47
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( 9 x + 8y
:
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,
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as carr e
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H R I ST S C O LL
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.
CA M B
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i
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A b o y o f en
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e
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s q are , t ree
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en it was o b s er e , t at
b er a
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o l
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s i e o f th e
eac
th e n
b er o f en
50
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a
on s
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(1)
2
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)
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.
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49 6
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.
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W OOLW I C H ]
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a
9
—
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9,
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t
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hu f h hd
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mil
74
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v k h im d w 4 mil
h fi
h u } mil h
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f m
v y h u I h w m y h u w uld h v k A ?
m
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b v (J
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M
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adratic I n x ,
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g x b et
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ith
th
s
Gi
.
+
my i
rat ic
o ri
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2
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mi
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th e
CO L L Ox
1
(
71
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m
v
d
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A p e rs o n in e s ts a c ertain s u
in a 6 p e r c en t G o e rn
lo an : if th e p ri c e h ad b een £3 les s h e o l
2; p e r
a e re c e i e
o re in te res t o n h is
o n ey ; at
at p ric e was t h e lo an is s e d ?
69
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in s ert e
rs t an d las t
2
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lo g b )
E
U
E
Q
[
ean s a re
If
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eq
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67
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24 1 7
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47
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at
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t to
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n
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[
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n
o
an
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t
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r;
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e
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o
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e
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es
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a
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N U
u l u mb
h
p v
M I S C E L LA
EO
Th e s erie s o f n at ra n
5, 6 , 7 , 8 , 9 ; an d S O o n : ro
3
3
r
o
i
s
7
t
n
1
(
) +
g
p
76
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S
77
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t
e
1
th e
at
su
mf
o
S E X A M PL E S
s o
l
__
3
e
n
o
f th e
up
u mb
to gro
m f th
m
ter
n
1
2
vd di
is di i
t at t h e s u
e rs
e
497
.
e n
1s e
q
e rs
s e ries
4
5
u
al
to
1
1
(2n
5 7
3
1)
In
2n
R
M
A
[
.
.
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.
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n
79
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v
th e
So l
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e
o
u
,
th e
In
e x p an s i o n
W OO LW I CH ]
.
o
.
f
—l
(
3
or
(
2
+ 1, 3
3
m+ 2
2x
1
2
—
1
x+x
.
at io n s
2
3
z
a
b
e
“
( I)
”
f mm m
f th e
eq
x
n
3
a cc o r
2, 3, 4 ;
in t h e
4
l
?
?
I
i
e
1
s
+y + z
x
(2)
x
v
LL
U
N
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[
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v u
Th e al e o f
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t o b e p o s iti e in tege rs
80
m
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xy z
h m
v
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in tegral
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v lu
a
n
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d b
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an
h
an
.
um
h
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hv
h
10 810 P _ 10 8 1 0 Q = P
84
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t
1 at n o
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+
9
p ers o n
an
a
rec e
a
e
an
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v
b e gi
n gs
s
n
S
a
s
s
Cx
.
.
]
a
s
t
e
e
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s o
°
m y w y my 20 hilli g
my iv l th 3 h illi
In h o w
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e
hm h v h
m f wh
wh
hw h
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t
n
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,
7 x — 3 , fin d th e
o s e lo garit
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b er o f in tegers
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a
t
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p
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83
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NC
TO
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I f ( x + ) is greater t
82
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t o 5 p e rs o n s
en
so
i
t
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u
m wi h i g h i tw d u gh t t
q
p
ml t dmt t
w
h th y m f g b q th d t th ld th
f
t i
m f m y i v t d t th tim f h i d t h 4 p
mul t d
t t k t 88 ;
h b qu th d th
u
d t th y u g
i t
t f
ml t h th f m b y £3 500 i v t d t th e m
m
t
f
t
h
t
t im i t h
3 p
t
t 63
Su p p i g t h i
g
m v td
th
t h i f th
d 1 4 wh t w
d th t h v b
17
A
85
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e
an
ca
e o
a c er a n s u
o
c en
n
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s
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o
or
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n
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e ac c u
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In
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wh
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a
e
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e ac
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as
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ne
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In
es
e
498
86
87
d
m
ers e
o r er :
m f mt m
v h
t
e
te
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rt
G E BR A
s c ale
d th e
n
.
wh
7
en e x p re s s e
u mb
J
N
S
[
i hm i l p g
er
n
.
T
)
(
S
.
m f th
J N
OH
—
x
)
2
’
C OL L
S
,y
—x
z
U
-
v
p
v
o s it i
m
m
m
5
1 +3 +
+
a
e a co
u
k
hu
H
to
n
o
a
rs
o
2
h
2
2
+ 19 2
+ 10 3
v h
t
e
d gre at er t
e an
h
.
1,
an
u
p
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it
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E
M
[ M
.
59 2
4 (9.
2_
(6
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9 3)
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ra
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ro
e o
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t
on
t t
.
ees e
o
9
CA M B
.
CA M B
.
]
x
p3
+ 9 3 = 03
2 ( 102 203 + a r.
es
1
4
n
te r
et
“
es t o n e
50
S
T
[
J N
an
a
OH
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C OL L
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A n A P,
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.
Te a, 23 6 d ; C o fi e e , l s
.
r ey s ,
— 2b
b
m
11 t
atio n s o
a
1 6 C lergy
2 58
th e p e r
a e
i
uk
.
.
+ b ro )
'
—6
or
26 3
.
hv
z
= 4,
u
(a w
3
,
557
.
‘
—4
or
2 p lac es
‘
6p r + 24 s
2q
(a w + b a )
,
le as t 3 7
at
S WE R S
P
§
=
7, or 3
y
N
AN D
.
S ON S , A T TH E
U N I VE R SIT Y PR E S S
.