MAC 1147
Review # 4
Sequences, Series, Binomial Theorem,
and Mathematical Induction
Prof. Nicoli-Suco
-------------------Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write out the first five terms of the sequence.
1) an = n - 4
A) -16, -12, -8, -4, 0
B) -3, -2, -1, 0, 1
C) -4, -3, -2, -1, 0
D) -1, 0, 1, 2, 3
B) -1, -4, -7, -10, -13
C) 1, 2, 3, 4, 5
D) 5, 8, 11, 14, 17
B
2) an = 3n - 2
A) 1, 4, 7, 10, 13
A
3) an = 4 n
A) 1, 4, 16, 64, 256
B) 4, 16, 64, 256, 1024
C) 1, 16, 81, 256, 625
D) 16, 64, 256, 1024, 4096
B
4) an =
A)
1
n2
1 2 3 4 5
, ,
,
,
4 9 16 25 36
C) 1,
1 1 1 1
, ,
,
4 9 16 25
D) 1,
2 3 4 5
, ,
,
4 9 16 25
B) 1, 1, 1, 1, 1
C) 0,
3 4 15 12
, ,
,
5 5 17 13
D) 1,
3
15
, 1,
,1
5
17
B) -1, 1, 3, 5, 7
C) -1, -1, 3, -5, 7
B) 1,
1 1 1 1
, , ,
2 3 4 5
C
n2 - 1
5) an =
n2 + 1
A) 0, 1 ,
4
13
,1,
5
12
C
6) an = (-1)n-1 (2n - 3)
A) 1, 1, -3, -5, -7
C
1
D) -1, -7, 3, -5, 13
Decide whether the given sequence is finite or infinite.
7) -5, -4, -3, -2
A) Finite
B) Infinite
A
8) -10, -9, -8, -7, ...
A) Infinite
B) Finite
A
Find the first six terms of the sequence.
9) a1 = -6, an = an-1 + 8
A) 2, 10, 18, 26, 34, 42
B) 0, 8, 16, 24, 32, 40
C) -6, 2, 10, 18, 26, 34
D) -6, 8, 16, 24, 32, 40
C
10) a1 = -3, an = 2 · an-1
A) 0, 2, -6, -4, -2, 0
B) -3, -6, -12, -24, -48, -96
C) -3, -6, -4, -2, 0, 2
D) -6, -12, -24, -48, -96, -192
B
11) a1 = 2, a2 = 2; for n
A) 2, 2, 4, 6, 8, 10
3, an = an-1 + an-2
B) 2, 2, 4, 4, 6, 6
C) 2, 2, 4, 6, 10, 16
D) 2, 2, 4, 8, 32, 256
B) 8
C) 30
D) 22
B) 162
C) 1080
D) 729
B) 20
C) 10
D) 17
C
Evaluate the sum.
4
12)
(k2 - 2)
k= 1
A) 14
D
6
13)
3k
k= 3
A) 19,683
C
14)
5
k=2
(k2 - 5)
2
A) 34
D
2
5
15)
i=2
(2i - 2)
A) 20
B) 12
C) 16
D) 18
B) 8.44
C) 11.55
D) 1409.97
B) 310
C) 4170
D) 3910
B) 297
C) 1147
D) 451
C) 18
D) -43
C) -15
D) -1
A
16)
5
i=2
9/i
A) -2.55
C
Use a graphing calculator to evaluate the series.
11
8j2 - 13
17)
j =2
A) 2186.67
D
18)
13
k2 - 5k + 7
k=3
A) 1301
D
Evaluate the sum using the given information.
19) x1 = -3, x2 = 4, x3 = 3, and x4 = -2
4
2
(- x i - 5)
i=1
A) -58
B) -50
A
20) x1 = -4, x2 = -1, x3 = 1, x4 = 3, and x5 = 0
5
(-2xi + 3)
i=1
A) 17
B) 5
A
Write the series using summation notation.
3
4
5
6
7
+
+
+
+
21)
1·2 2·3 3·4
4·5
5·6
A)
5
i=1
i+2
i(i - 1)
B)
5
i=1
i
i(i + 1)
C)
5
i=1
D
3
i-2
i(i + 1)
D)
5
i=1
i+2
i(i + 1)
22) 16 + 26 + 3 6 + 4 6 + . . .
6k
A)
B)
k=0
4
k6
k6
C)
k=1
k5
D)
k=1
k=1
C
Find the common difference for the arithmetic sequence.
23) 2, 3, 4, 5, ...
A) 0.01
B) -1
C) 1
D) 3
B) -1
C) -2.6
D) -3
C
24) -9, -10, -11, -12, ...
A) -2
B
Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
25) The first term is 17, and the common difference is 3; n = 5
A) 17, 3, 23, 26, 29
B) 3, 20, 37, 54, 71
C) 17, 20, 23, 26, 29
D) 17, 3, 40, 60, 80
C
26) The first term is -1 +
17, and the common difference is 3; n = 3
A) -1 +
17, 2 +
17, 5 +
17
B) -1 +
C) -1 -
17, 2 +
17, 5 +
17
D) 1 +
17, 3 +
17, 2 +
17, 6 +
17, 5 +
17
17
A
Find a
n
and a
6
for the following arithmetic sequence.
27) 4, 10, 16, 22, 28, ...
A) an = 2(3n - 1); a6 = 34
B) an = 2n - 6; a6 = 6
C) an = 4(6)n-1 ; a6 = 24
D) an = 6n - 1; a6 = 35
A
28) -1, 1, 3, 5, 7, ...
A) an = 3n - 2; a6 = 16
B) an = n + 2; a6 = 8
C) an = 2n - 3; a6 = 9
D) an = -1(2)n-1; a6 = -32
C
29) a1 = -36, d = 5
A) an = -36 - 5(n - 1), a 6 = -61
B) an = -36 + 5n, a6 = -6
C) an = -36 - 5n, a6 = -66
D) an = -36 + 5(n - 1), a 6 = -11
D
4
Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest
hundredth.
30) a
15
= 320, a
A) a
1
69
= 1400
= 40, d = 1080
B) a
1
= 1080, d = 20
C) a
= -5, d = 4
C) a
1
= 40, d = 20
D) a
= -5, d = 132
D) a
1
= 1080, d = 40
C
31) a
7
= 19, a
A) a
1
40
= 151
= 132, d = -5
B) a
1
1
1
= 132, d = 4
B
32) S3 = 9, a3 = 5
A) a1 = 1, d = 2
B) a1 = 1, d = 3.33
C) a1 = -9, d = 3.33
D) a1 = -9, d = 2
A
Find the nth term of the geometric sequence.
33) a1 = 5, r = 3, n = 4
A) a4 = 3375
5
27
B) a4 = 27
C) a4 =
B) a11 = 177,147
C) a11 = -99
B) a4 = 30
C) a4 =
1
30
D) a4 = -30
1
B) a5 =
2048
C) a5 =
1
512
1
D) a5 =
64
D) a4 = 135
D
34) a1 = 3, r = -3, n = 11
A) a11 = -59,049
D) a11 = 531,441
B
1
35) a1 = 1920, r = , n = 4
4
A) a4 = -
1
30
B
36) a1 =
1
1
,r= ,n=5
2
4
A) a5 =
1
128
C
Find a general term an for the geometric sequence.
37) a1 = 4, r = 4
A) an =4( 4)(n-1)
B) an = 4 n-1 + 3
C) an = 4(4)n-1
C
5
D) an = 4 + 12(n - 1)
1
38) a1 = , r = 6
3
A) an =
1 n-1
·6
3
B) an =
1 5
+ (n - 1)
3 3
C) an =
1
+ 6(n - 1)
3
D) an =
1 n-1 5
+
3
3
A
39) a1 = 8, r =
8
2
A) an = 8 n-1 - 4
n-1
1
B) an = 8 + (n - 1)
2
C) an = 8 · 4
D) an = 8 - 4(n - 1)
C
Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
40) a2 = 12, a4 = 108
A) a1 = 12, r = 3
B) a1 = 4, r = 0.33
C) a1 = 108, r = 0.33
D) a1 = 4, r = 3
B) a1 = 16, r = 2
C) a1 = 8, r = 2
D) a1 = 256, r = 0.5
D
41) a2 = 64, a5 = 8
A) a1 = 128, r = 0.5
A
Use the formula for Sn to find the sum of the first five terms of the geometric sequence.
42) 4, 8, 16, 32, . . .
A) 134
B) 124
C) 122
D) 126
B) 682
C) -682
D) -410
B) 1
C) 9
D) 36
B) 120
C) 60
D) 720
B
43) 2, -8, 32, -128, . . .
A) 410
A
Evaluate the expression.
9!
44)
7! 2!
A) 0!
D
45)
10
7
A) 1
B
6
8
46) 8
A) 1
B) 2
C) 40,320
D) 0
B) 126
C) 30,240
D) 252
A
47)
10!
5!5!
A) 504
D
Write the binomial expansion of the expression.
48) (2x + 3)3
A) 4x6 + 6x3 + 729
B) 4x2 + 12x + 9
C) 8x3 + 36x2 + 54x + 27
D) 8x3 + 36x2 + 36x + 27
C
49)
3
1
x+2
3
A)
1 3 2 2
x + x + 4x + 8
9
3
B)
1 3 2 2
x + x + 4x + 8
27
3
C)
1 3 2 2
x + x +8
9
3
D)
1 6 4 3
x + x +8
27
9
B
50) (3x - 2)4
A) -162x4 + 432 x3 + 216x 2 + 192x + 16
B) 81x4 - 216x3 + 216x2 - 96x + 16
C) (9x2 - 6x + 4)4
D) 81x3 - 216x2 + 216x - 96
B
51) (3x - 2)5
A) 243x5 + 240x4 - 720x3 - 720x2 + 240x - 32
B) 243x5 - 810x4 + 1080x3 - 720x2 + 240x - 32
C) (9x2 - 12x + 4)5
D) 243x5 - 162x4 + 108x3 - 72x 2 + 48x - 32
B
Write the indicated term of the binomial expansion.
52) (6x + 9)3 ; 3rd term
A) 81
B) 2916x
C) 972x2
D) 1458x
B) -108,864x5 y4
C) -326,592x4 y5
D) 163,296x4 y6
D
53) (3x - 2y)9 ; 6th term
A) 163,296x5 y4
C
7
54) (x - 2y)10; 8th term
A) -15,360x7 y3
B) 7680x7 y3
C) -15,360x3 y7
D) 7680x3 y8
B) 5120
C) 8000x2
D) 1600x
B) 240x
C) 192
D) 300x2
C
55) (5x + 4)5 ; 5th term
A) 6400x
A
56) (5x + 4)3 ; 3rd term
A) 960x
B
Use mathematical induction to prove that the statement is true for every positive integer n.
57) 6 + 12 + 18 + ... + 6n = 3n(n + 1)
58) 12 + 42 + 72 + . . . + (3n - 2)2 =
n(6n2 - 3n - 1)
2
59) 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) =
60) 1 -
1
2
1-
n(n + 1)(n + 2)
3
1
1
1
... 1=
3
n+1
n+1
8