Precalculus
UNIT S&S
SEQUENCES
SERIES
AND
REVIEW
S&S-1
1)
2)
SEQUENCES
Identify each of the following sequences as either arithmetic, geometric, or neither.
Write an explicit definition for each one.
a)
5, -20, 80, -320, …..
c)
900, 1200, 1600,
e)
86, 59, 32, 5, …..
3
, …..
64, 77, 90, 103, …..
d)
1 8 27 64
, ,
,
, .....
2 3 4 5
f)
-1 2 -3 4
, ,
, , .....
3 5 7 9
Determine the indicated term for the given sequence.
a)
t26 for 56, 28, 14, 7, …..
b)
t47 for 11, 33, 99, 297, …..
c)
t213 for 7, -3, -13, -23, …..
d)
t40 for 28, 39, 50, 61, …..
e)
Arithmetic sequence
t1 for t7 = 63 and t12 = 28
f)
Arithmetic sequence
t51 for t11 = 16 and t18 = 72
g)
Geometric sequence
h)
Geometric sequence
t12 for t1 =
3)
6400
b)
1
27
and t4 = 1
t15 for t5 = 2000 and t8 = -250
Find the next four terms for each of the following sequences.
a)
t1 = 21
tn = tn - 1 + 3n
c)
t1 = 8 t2 = -5
tn = tn -1 + tn - 2
MPH/CR 5/09
b)
t1 = 6
tn = n  tn – 1 - 11
1
Precalculus
4)
Write a recursive definition for each of the following sequences.
a)
24, 19, 14, 9, …..
b)
324, -216, 144, -96, …..
c)
4, 13, 40, 121, …..
d)
7, 4, 11, 15, 26, 41, …..
S&S-2
5)
FINITE SERIES
Identify each of the following series as either arithmetic or geometric when not
indicated. Find the indicated sum for each.
a)
Arithmetic series
S15 for t1 = -6 and t15 = -45
b)
Arithmetic series
S87 for tn = 5 – 6n
c)
Geometric series
d)
Geometric series
æ 1ö
S11 for tn = 286 ç ÷
è 2ø
n
S23 for t1 = 3 and r = -2
e)
26 + 33 + 40 + 47 + ….. + 404
f)
S14 for 54 – 18 + 6 – 2 + …..
g)
S75 for 93 + 77 + 61 + 45 + …..
h)
S12 for 2 2 + 4 + 4 2 + 8 + .....
S&S-3
LIMITS
OF
INFINITE SEQUENCES
6)
Evaluate each of the following limits if possible.
a)
lim
d)
lim
n®¥
n®¥
9
n2
6n3
1 - 8n
MPH/CR 5/09
b)
e)
lim - 8n3
n®¥
lim (-1) n
n®¥
6 - 14n2
c)
lim
f)
æ 1ö
lim cos ç ÷
n®¥
è nø
n®¥
22n2 + n
2
Precalculus
g)
æ -2 ö
lim ç ÷
n®¥ è 5 ø
j)
æ 8ö
lim ç ÷
n®¥ è 3 ø
S&S-4
7)
n
h)
lim - 16
k)
lim tan
n®¥
n
n®¥
np
2
n
i)
lim log 10
l)
lim
n®¥
15n2 - 3n
n®¥
n2 + 5
INFINITE SERIES
Determine whether the following series converge or diverge. If the series
converges, find its sum.
9
27
81
a)
1 - 3 + 9 - 27 + ……
b)
6+
c)
28 + 17 + 6 - 5 - ……
d)
-12 + 18 - 27 +
e)
11 + 15 + 19 + 23 + …..
f)
96 - 48 + 24 - 12 + …...
2
+
8
+
32
+ .....
81
2
- …..
8)
Find the common ratio for an infinite geometric series with t1 = 14 that converges
to 18.
9)
What is the value of x if 4 + 8x + 16x2 + 32x3 + ….. converges to 6?
10) Find the interval of convergence for each of the following series. Express the sum
of each series in terms of x.
a)
3 – 12 (x - 7) + 48 (x - 7)2 – 192 (x - 7)3 + …..
b)
1+
MPH/CR 5/09
3x
2
+
9x2
4
+
27x3
8
+ .....
3
Precalculus
11) Express each of the following as a rational number.
a)
S&S-5
0.429429429…..
b)
11.324324324…..
SIGMA NOTATION
12) Expand and evaluate each of the following.
a)
7
å -2k (5k + 3)
b)
k=2
5
å 8m
3
+ 11
m=1
13) Evaluate each of the following.
a)
50
å 4(2)
b)
k
k=-2
c)
29
å (7n - 2)
n= 4
29
å (2k + 1)(3k - 7)
d)
k=1
43
å (m
3
+ 9m + 2)
m=1
14) Express each of the following using sigma notation.
1
1
1
1+
c)
log 4x + log 5x + log 6x + log 7x + ……… + log 34x
d)
f)
1
18
4
+
+
1
6
9
+
+
1
2
16
+ ..... +
1
a)
+ ..... +
49
81
2
sin 4x + sin 8x + sin 12x + …..
MPH/CR 5/09
b)
e)
g)
7 + 3 - 1 - 5 - 9 - 13 - …..
-1
2
+
3
22x
1
4
+
-
1
6
5
22x
+
+
1
8
- .....
7
22x
+ ...... +
39
22x
4
Precalculus
APPLICATIONS
OF
SEQUENCES
AND
SERIES
Answer each of the following word problems. Make sure you distinguish between
sequence and series problems and arithmetic and geometric problems.
15) How many three-digit numbers are multiples of fifteen?
16) Find the sum of the set of all the four-digit numbers divisible by 18.
17) You are building a savings account. You save $40 the first month, $52 the next, $64
the third month, and so on. What is the total amount of money you will have saved
after two years?
18) A biology experiment begins with 27 flobdoodles. Each flobdoodle reproduces only
once, and in that reproduction produces 3 flobdoodles. Hence, after one
reproduction cycle, the original 27 flobdoodles are alive and have been joined by 81
offspring. These 81 flobdoodles then produce 243 additional offspring, and the
process continues. After fourteen reproduction cycles, have many flobdoodles are
there?
19) A ball is dropped from a height of 72 feet and rebounds three-fourths of the way.
How far will the ball travel before coming to rest?
20) A landscape design artist is constructing a brick pyramid in Mrs. Fibonacci’s
backyard. The base has 47 bricks, and each successive base has 4 less bricks. The
last row has 3 bricks. How many bricks are in the pyramid design?
21) Archie is saving money toward his car insurance bill. The first week he saves $2,
the next week $4, and the next week $8. If he continues at that rate, how much
will he have saved after ten weeks? Would it be enough to pay his premium of
$2700?
MPH/CR 5/09
5
Precalculus
ANSWERS
1)
2)
3)
4)
5)
6)
7)
a)
Geometric tn = 5 (-4)n-1
b)
Arithmetic tn = 13n + 51
d)
Neither tn =
f)
Neither tn =
n-1
c)
æ 4ö
Geometric tn = 900 ç ÷
è 3ø
e)
Arithmetic tn = 113 – 27n
a)
æ 1ö
56 ç ÷
è 2ø
e)
105
a)
21, 27, 36, 48, 63
c)
8, -5, 3, -2, 1, -1
a)
n3
n+1
(-1) n n
2n + 1
25
b)
11(3)46
c)
-2113
d)
f)
336
g)
6561
h)
b)
6, 1, -8, -43, -226
t1 = 24; tn = tn-1 – 5
b)
t1 = 324; tn =
c)
t1 = 4; tn = 3tn-1 + 1
d)
a)
-382.5
b)
-22533
c)
e)
11825
f)
40.49999153
g)
a)
0
b)
-
c)
e)
DNE
f)
1
g)
i)
0
j)

a)
diverges
b)
24
e)
diverges
f)
64
MPH/CR 5/09
457
125
64
-2
tn-1
3
t1 = 7; t2 = 4; tn = tn-1 + tn-2
292721
d)
8388609
h)
252 + 126 2
d)
-
0
h)
-16
k)
DNE
l)
c)
diverges
d)
1024
-37425
-7
11
15
diverges
6
Precalculus
8)
2
9)
9
10) b)
-2
3
<x<
2
3
; S =
1
6
2
2 - 3x
12) a)
-1552
b)
1811
13) a)
253 – 1
b)
2951
14) a)
7
å
n=1
e)
¥
å
n=1
1
n
2
(-1) n
2n
b)
¥
å (11 - 4n)
10a)
11a)
c)
c)
n=1
f)
¥
å sin 4nx
n=1
27
4
<x<
143
333
46342
34
å log nx
29
4
; S =
3
4x - 27
b)
d)
d)
n= 4
g)
¥
å
n=1
419
37
903516
6
å
n=1
1
18
(3) n-1
2n + 1
22x
15) Arithmetic Sequence 60
16) Arithmetic Series 2749500
17) Arithmetic Series $4272
18) Geometric Series 64570068
19) Geometric Series 504 ft
20) Arithmetic Series 300
21) $2046 – It would take 11 weeks to accumulate the required funds.
MPH/CR 5/09
7