Chapter 14
Sequences, Series, and the Binomial Theorem
14.1 Check Points
1. a.
1
25
2(1) 5 7
n
an
a
b. (1)
21
n
nn
a
1
11
(1) 1 1
33
a
2.
1
20
(1)!
20 20 10
(1 1)! 2!
n
an
a
3. a.
6
2
2
i
b.
5
3
345
23
23 23 23
k
k
c.
5
1
4 44444 20
i
4. a. The sum has nine terms, each of the form
2
i,
1
i
14.1 Concept and Vocabulary Check
5. factorial; 5; 1; 1
6.
1
a
;
2
a
;
3
a
;
n
a
; index; upper limit; lower limit
Chapter 14 Sequences, Series, and the Binomial Theorem
14.1 Exercise Set
1.
32
n
an
2.
1
41 1 4 1 3
a
42 1 8 1 7
a
3.
3
n
n
a
1
1
33
a
2
2
39
a
3
3
327
a
4
4
381
a
The first four terms are 3, 9, 27, 81.
4.
1
1
11
33
a
5.
3
n
n
a
6.
1
1
11
33
a
2
7.
13
n
n
an
4
4
14317 7a
The first four terms are –4, 5, –6, 7.
8.
11 2
1
114 15155a
21 3
2
124 16166a
31 4
3
134 17177a
41 5
4
144 18188a
34 7
4
24 81
44 8
a
Section 14.1 Sequences and Summation Notation
10.
1
31 31
15 6 2
a
45 9 3
11.
1
1
n
nn
a
12.
11 2
11
11
1
21 3
21
a
21 3
22
11
11
41 5 5
21
a
31 4
33
11
1
81 9
21
a
13.
2
n
n
an
3
3! 3 2 1 2
2
4
4162
4! 4 3 2 1 3
a
15.
1
2
3
4
2( 1)!
2(1 1)! 2 2! 2 2 1 4
2(2 1)! 2 3! 2 3 2 1 12
2(3 1)! 24! 24321
48
2(4 1)! 2 5! 2 5 4 3 2 1
240
n
an
a
a
a
a
26 12
Chapter 14 Sequences, Series, and the Binomial Theorem
17.
6
1
5 5 1 5 2 5 3 5 4 5 5 5 6 5 10 15 20 25 30 105
i
i
21.
5
1
4 11 4 22 4 33 4 44 4 55 4
15 26 37 48 59 5 12 21 32 45 115
k
kk
22.
4
1
322314051664064
k
kk
24.
234
4
2
1 1 1 1111191319317
3 3 3 3 9 27 81 9 9 27 3 81 81 81 81 81
i
i
25.
9
5
11 11 11 11 11 11 55
i
Section 14.1 Sequences and Summation Notation
28.
1 0111213141 12 34 5
4
0
1 1 1 1 1 1 11111
1! 0 1! 1 1! 2 1! 3 1! 4 1! 1! 2! 3! 4! 5!
11 1 1 1 111 1
1
121321432154321 2624120
120 1 60 1 20 1 5 1 120 60
1120 2 60 6 20 24 5 120 120 1
i
i
i
20 5 1 76 19
20 120 120 120 120 30
30.
5
1
2! 1 2! 2 2! 3 2! 4 2! 5 2! 3! 4! 5! 6! 7!
!1!2!3!4!5!1!2!3!4!5!
321 432! 543! 654! 765! 43 2!
6
12!3!4!5!
i
i
i
2!
54 3!
3!
65 4!
4!
76 5!
5!
612203042 110
1
i
34.
12
23 12
1
55 5 ...5 5
i
i
35.
30
1
1 2 3 … 30
i
i
36.
40
1
1 2 3 … 40
i
i
Chapter 14 Sequences, Series, and the Binomial Theorem
41.
1
135... 2 1 2 1
n
i
ni
44.
16
3
6 8 10 12 ... 32 2
k
k
45.
12
212
0
…
k
k
aarar ar ar
48.
2
1
…
n
nk
k
ad ad ad ad
Section 14.1 Sequences and Summation Notation
51.
5
1
(2 ) 2( 4) 4 2( 2) 2 2(0) 0 2(2) ( 2) 2(4) ( 4)
4(2)024 0
ii
i
ab
5
4
i
55.
55
2 2 2 2222 222 2 2
11
(4)(2)024 420(2)(4)
1640416 1640416
80
ii
ii
ab
57. a.
7
1
11
2.7 3.0 3.2 3.7 4.4 4.9 5.3 3.9
77
i
i
a
For the time period shown by the graph, U.S. adult users spent an average of 3.9 hours per day on digital media.
b.
7
1
11
2.5 3.0 3.5 4.0 4.5 5.0 5.5 4
77
i
i
a
The model overestimates the actual average by 0.1 hour.
59.
20 20 20
20
0.06
6000 1 6000 1 0.015 6000 1.015 8081.13
4
a
The balance in the account after 5 years if $8081.13.
Chapter 14 Sequences, Series, and the Binomial Theorem
66.
5
5
3 243
24000 24000 5695.31
4 1024
a
After 5 years, the car is worth $5695.31; At the end of each year, the car’s value is 3
4 of its value from the previous
year.
70. 100
n
an
;
As n gets larger, the terms get closer to 0.
71.
2
3
257
n
nn
an
As n gets larger, the terms get closer to 0.
74. does not make sense; Explanations will vary. Sample explanation: Any of the terms of this sequence could be negative
and/or include non-integers.
75. makes sense
Section 14.1 Sequences and Summation Notation
78. false; Changes to make the statement true will vary. A sample change is:
222
111
ii i i
iii
ab a b
79. true
80. false; Changes to make the statement true will vary. A sample change is:
67
22
11 1
ij
ij
14916253649 28
81.
1
n
an
82.
2
n
an
83.
1
n
n
a
84.
(2)
n
ann
Chapter 14 Sequences, Series, and the Binomial Theorem
90.
2
(4)! (3)(4) 712
(2)!
nnn nn
n
92.
4
1
log 2 1 log 2 2 log 2 3 log 2 4
log 2 log 4 log 6 log8 log 384
log 2
i
i
94.
1
1
221 1
331 2
441 3
5
7
557512
5512517
5517522
nn
aa
a
aa a
aa a
aa a
The first four terms are 7, 12, 17, 22.
97.
66 5
22 22
22
66 5
22
xx xx
xx
xx
560 or 40
56 4
xx
xx
32
43
7(2) 5aa
54
12 ( 7) 5aa
The difference between consecutive terms is always
5.
100.
8
4( 1)(7)
4 (8 1)( 7) 4 (7)( 7) 4 49 45
n
an
a
14.2 Check Points
1.
1
6
100
20 ( 30) 50
a
a
2.
1
6, 5ad
To find the ninth term,
9
a, replace n in the formula
Section 14.2 Arithmetic Sequences
3. a.
1
(1)
16 ( 1)0.35
n
aand
n
4. 3, 6, 9, 12, …
To find the sum of the first 15 terms,
15
S
, replace n in the formula with 15.
3.
1
(1)
n
aand
5.
30
1
(6 11) (6 1 11) (6 2 11) (6 3 11) (6 30 11) 5 1 7 169
i
i
6.
1800 64,130
n
an
1
1800(1) 64,130 65,930a
Chapter 14 Sequences, Series, and the Binomial Theorem
14.2 Concept and Vocabulary Check
1. arithmetic; common difference
14.2 Exercise Set
1. Since 6 – 2 = 4,
4.d
2. Since
835, 5.d
7.
1
200
200 20 220
a
a
8.
1
300a
2
300 50 350a
9.
1
2
7
74 3
a
a
3
4
5
6
257
7512
12 5 17
a
a
a
11.
1
2
300
300 90 210
a
a
3
4
140 60 80
80 60 20
a
a
2
51 42
222
41 3
a
5
6
122
110
22
a
a
Section 14.2 Arithmetic Sequences
14.
1
3
4
a
15.
1
2
3
0.4
0.4 1.6 2
21.6 3.6
a
a
a
16.
1
2
0.3
0.3 1.7 2
a
a
17.
6
13 6 1 4 13 5 4
13 20 33
a
18.
16
9 16129 15293039a
21.
200
40 200 1 5 40 199 5
40 995 955
a
24.
70
32 70 1 4 32 69 4
32 276 244
a
27.
1
17 14
74 4114
n
aan d n
nn
20
11 4 20 11 80 69a
29.
1
12014
20 4 4 4 16
n
aan d n
nn
70 5 5 5 65
nn
20
5 20 65 100 65 165a
31.
1
11
11
33
11 1 1 2
n
aan d n
32.
1
11
11
44
11 1 1 2 1 1
n
aan d n
Chapter 14 Sequences, Series, and the Binomial Theorem
35. First find
20
.a
20
420164196
4 114 118
a
20
20 4 118 10 122 1220
2
S
37. First find
50
.a
50
10 50 1 4 10 49 4
10 196 186
a
39. First find
100
.a
60
60 2 120 30 122 3660
2
S
42.
80
2 80122 7922158160a
80
80 2 160 40 162 6480
2
S
11 1
12
n
n
12
12 22 44 6 66 396
2
S
Section 14.2 Arithmetic Sequences
45.
17
53 513 523 533...5173
i
46.
20
1
64 614 624 634...6204
6 4 12 4 18 4 … 120 4 2 8 14 … 116
i
i
20
20 2 116 10 118 1180
2
S
2
48.
40
1
2 6 2 1 6 2 2 6 2 3 6 … 2 40 6
26 46 66 ... 806 420... 74
i
i
40
40 4742070 1400
2
S
50.
50
1
4 4 1 4 2 4 3 … 4 50 4 8 12 … 200
i
i
Chapter 14 Sequences, Series, and the Binomial Theorem
52. First find
16
a
and
18
b
:
16 1
(1)
aand
53.
1
1
83 1 ( 1)( 3 1)
83 1 4( 1)
84 4 4
88 4
22
n
aan d
n
n
n
n
n
There are 22 terms.
54.
1
1
n
bbn d
55.
1
2
nn
n
Saa
For
{}:
n
a
14 1 14
14 71 ( 51) 350
Saa
56. First find
15
a
and
15
b
:
15 1
(1)
aand
15 1 15
15 7.5 1 ( 55)
2
405
Saa
And then for
{}:
n
b
15 1 15
15 7.5 3 73 570
2
Sbb
So,
15 15
11
570 ( 405) 975
nn
nn
ba
22
()
14(1)
144
45
yy mxx
yx
yx
yx
Section 14.2 Arithmetic Sequences
59. Using
1
(1)
n
aand
and
2
4:a
21
1
(2 1)
4
aa d
ad
1
Substituting the value into the second equation and
solving for d:
16 (4 ) 5
dd
60. Using
1
(1)
n
aand
and
3
7:
a
31
1
(3 1)
72
aa d
ad
Substituting the value into the second equation and
solving for d:
17 (7 2 ) 7
17 7 5
10 5
dd
d
d
18.4 ( 1)0.6
18.4 0.6 0.6
0.6 17.8
n
an
n
n
24.4 0.3 0.3
0.3 24.1
n
n
b.
0.3 24.1
0.3(30) 24.1
n
an
Chapter 14 Sequences, Series, and the Binomial Theorem
Company B
64. Company A:
23000 1 1200
23000 1200 1200 1200 21800
n
an
nn
65. a. Total cost:
$4565 $4824 $5046 $5241 $19,676
b.
1
225(1) 4357 4582
a
66. a. Total cost:
$5249 $5413 $5629 $5837 $22,128
b.
1
198(1) 5037 5235
a
67. Answers will vary.
442,500
The total salary over a ten-year period is $442,500.
10
2
5(61, 400) $307,000
S
Company B
110
2
n
n
Saa
Section 14.2 Arithmetic Sequences
70.
26
30 26 1 2
30 25 2
a
There are 1430 seats in the theater.
71.
38 1
(1)
20 (38 1)(3)
aand
72. – 77. Answers will vary.
78. does not make sense; Explanations will vary.
Sample explanation: The difference between terms
is not constant. Thus, this is not an arithmetic
sequence.
79. makes sense
86. Answers will vary.
For example, consider
42 1
an
.
87. From the sequence, we see that
1
21700a
and
23172 21700 1472.d
314,628 is the 200
th
term of the sequence.
88.
10
23 10 1 2
23 9 2
23 18
41
a
nn n
Chapter 14 Sequences, Series, and the Binomial Theorem
90.
2
log( 25) log( 5) 3xx
91.
2
310xx
Solve the related quadratic equation.
2
3100
xx
The boundary points are
5 and 2.
Interval Test Value Test Conclusion
,5
−6
2
63610
18 10, false
,5
does not belong to the solution set.
92.
Pt
APt
AP t Pt