Sequence and Series
Sequence is a set of quantities arranged in a definite order.
1, 2, 3, 4, 5,…
-1, 2, -4, 8, -16…. 16, 12, 8, 4, 0, -4…..
Arithmetic Sequences
Arithmetic sequences (or arithmetic progressions) have a common difference d
between terms
First term labeled as a
The terms of a sequence are generally labeled
u1, u2, u3,……un
The “nth term” of a sequence is labeled un
un = un-1 + d
u1, u1+d, u1+2d….u1+(n-1)d
General term is un = a+(n-1)d
Arithmetic Series
Series is when the terms of a sequence are added.
1+2+3+4+5…. 16+12+8+4+0-4
Sn Is the sum of n terms of a series
Sn = u1 + u2 + u3 + .... + un
Sn = a + (a + d) + (a + 2d) + ....(a + (n - 3)d) + (a + (n - 2)d) + (a + (n -1)d)
Sn = u1 + u2 + u3 + .... + un-2 + un-1 + un
Sn = un + un-1 + un-2 + ... + u3 + u2 + u1
Sn + Sn = (u1 + un ) + (u2 + un-1 ) + ... + (un + u1 )
2Sn = éëu1 + u1 + ( n - 1) d ùû + éëu1 + u1 + ( n - 1) d ùû + ... + éëu1 + u1 + ( n - 1) d ùû
2Sn = éë 2u1 + ( n - 1) d ùû + éë 2u1 + ( n - 1) d ùû + ... + éë 2u1 + ( n - 1) d ùû
2Sn = n éë 2u1 + ( n - 1) d ùû
n
\Sn = éë 2u1 + ( n - 1) d ùû
2
Ex.1
The second term of arithmetic sequence if 7. The sum of the first four terms of
the arithmetic sequence is 12. Find the first term, a, and the common difference,
d, of the sequence.
u2 = 7 = u1 + ( 2 - 1) d = u1 + d
4
[ 2u1 + 3d ]
2
6 = 2u1 + 3d
(14 = 2u1 + 2d) - (6 = 2u1 + 3d)
\d = -8
7 = u1 - 8
\u1 = 15
S4 = 12 =
Sigma Notation
∑ is “the sum of”…
n
åu
i
= u1 + u2 + u3 + ... + un-1 + un
i=1
Sigma Properties
n
n
i =1
i =1
å ai = aå i
n
å b = nb
i =1
n
n
n
n
n
n
i =1
i =1
i =1
i =1
i =1
i =1
å (ai + b) = å ai + å b = aå i + å b = aå i + nb
Ex. 2
3
å (2i + 3) = 5 + 7 + 9 = 21
i =1
3
3
3
i =1
i =1
i =1
å 2i + å 3 = 2å i + (3 * 3) = 2(1 + 2 + 3) + 9 = 2(6) + 9 = 12 + 9 = 21
Geometric Sequences
A sequence of numbers in which the ratio between consecutive terms is always
the same
un = un-1 * r where r is the common ratio
General Term
un = ar n-1 where r is the common ratio and a is the first term
Geometric Series
Sn = u1 + u1r + u1r 2 + ... + u1r n - 3 + u1r n - 2 + u1r n -1
rSn = u1r + u1r 2 + ... + u1r n - 3 + u1r n - 2 + u1r n -1 + u1r n
Sn - rSn = u1 - u1r n
Sn (1 - r) = u1 (1 - r n )
\ Sn =
u1 (1 - r n )
,r < 1
(1 - r)
rSn - Sn = u1r n - u1
Sn (r - 1) = u1 (r n - 1)
\ Sn =
u1 (r n - 1)
,r > 1
(r - 1)
Sum to infinity when r < 1
u1 (1 - r n )
Sn =
,r < 1
(1 - r)
¥
æ 1ö
çè ÷ø ® 0
2
rn ® 0
u (1 - 0)
S¥ = 1
(1 - r)
u1
\ S¥ =
,r < 1
(1 - r)
Ex. 2
The sum of an infinite geometric sequence is 13.5, and the sum of the first three
terms is 13.
Find the first term.
u1
1- r
u (1 - r 3 )
S3 = 13 = 1
1- r
13(1 - r)
u (1 - r 3 )
= 1
13.5(1 - r)
u1
13
= 1 - r3
13.5
13
r3 = 1 13.5
1
r=
3
u
13.5 = 1
2
3
\u1 = 9
S¥ = 13.5 =
Ex. 3
For the geometric sequence 3, 12, 48…, find the common ratio, the “nth term”, and
the first term exceeding 1000.
u1 = 3
12
=4
3
un = 3(4)n -1
r=
un > 1000
3(4)n-1 > 1000
1000
4 n -1 >
3
æ 1000 ö
log(4 n-1 ) > log ç
è 3 ÷ø
æ 1000 ö
(n - 1)log 4 > log ç
è 3 ÷ø
æ 1000 ö
log ç
è 3 ÷ø
n>
+ 1 = 5.19
log 4
n=6
for when the first term exceeds 1000