Chapter 14 Review Exercises
16.
200 1 20
200 20 20
n
an
n
17.
1
12 1 2
n
an
18.
815 7
15 1 7 15 7 7
n
d
an n
19. First, find d.
12 5 7d
20. First, find d.
363d
Next, find
15
.a
Chapter 14 Sequences, Series, and the Binomial Theorem
21. We are given that
100
300a
,
1
3a
, and
100n
.
22.
16
3 2 31 2 32 2 33 2 ... 316 2
i
23.
25
1
26 216 226 236... 2256
i
i
24.
30
1
5515253...530
i
i
25. a.
40.36 1 1.63
40.36 1.63 1.63
1.63 38.73
n
an
n
n
26.
10
31500 10 1 2300
31500 9 2300
31500 20700 52200
a
Chapter 14 Review Exercises
27.
35
25 35 1 1 25 34 1
25 34 59
a
28. The first term is 3.
The second term is
32 6
.
29. The first term is
1
2
.
The second term is
11 1
22 4
.
30. The first term is
16
.
The second term is
1
16 4
4
.
31. The first term is –5.
The second term is
515
.
The third term is
51 5
.
The fourth term is
515
.
The fifth term is
51 5
.
35.
1
1
1
81 7
8
12
12 12 1128 128
n
n
n
aar
a
81 7
8
11
100 100
10 10
a
37.
1
1
1
81 7
41
12 3
1
12 3
n
n
n
d
aar
1
5
15
51 3 5 1 14348907
S
39.
1
82
ra
7
7
11
81 81
2128
11
122
S
Chapter 14 Sequences, Series, and the Binomial Theorem
40.
6
6
51 5 5 1 15625
515 4
i
42.
5
1
5
1
11
21 21
4
11024
23
1
414
4
1023
21023 4 341
1024
3512 3 128
4
i
i
44.
2
1
11
22
a
ra
45.
2
1
42
63
a
ra
665
6
5
23
13
3
S
47.
1
66
69
10 10
0.6 19
a
49. a. Divide each value by the previous value:
18.80 1.02
18.44
18.44 1.02
18.15
18.15 1.02
17.86
17.86 1.02
17.58
16.76 1.02
16.50
16.50 1.02
b.
1
15.98(1.02)
n
n
a
c. 2080 is 8 decades after 2000 so n = 8.
1
31 1
15.98(1.02)
n
n
a
Chapter 14 Review Exercises
50.
1.06r
1
1
1
32000 1.06
n
n
n
aar
51. a.
11
nt
r
n
r
n
P
A
$520, 0.06, 1, 20
Prnt
$8729
52. a.
11
nt
r
n
r
n
P
A
12 30
100, 0.055, 12, 30
0.055
$100 1 1
12 $91,361
Pr nt
Chapter 14 Sequences, Series, and the Binomial Theorem
53.
70% 0.7r
54.
11 11! 11! 11 10 9 8!
88! 11 8 ! 8!3!
8! 165
3!
56. Applying the Binomial Theorem to
3
21x
, we have
2 , 1, and 3.axb n
57. Applying the Binomial Theorem to
4
2
1x
, we have
2
, 1, and 4.axb n
Chapter 14 Review Exercises
58. Applying the Binomial Theorem to
5
2xy
, we have , 2 , and 5.axb y n
59. Applying the Binomial Theorem to
6
2x
, we have
, 2, and 6.axb n
23456
65 4 3 2
6
66 6 6 6 6 6
22 2 222
01 2 3 4 5 6
2
xxxxxx
x
60.
8
2
3x
61.
9
3x
22! 9 2 !
r
21 7!
Chapter 14 Sequences, Series, and the Binomial Theorem
62.
5
2x
63.
6
23x
Chapter 14 Test
1.
1
2
1
n
n
an
11 2
12
11
11
11
1
a
2.
5
222222
10 1 10 2 10 3 10 4 10 5 10
i
4.
12
945
41545551
512 1 60 1 59
n
d
an nn
a
Chapter 14 Test
6. First, find d.
14 ( 7) 7d
7.
20
1
34 314 324 334...3204
34 64 94 ... 604
125...56
i
i
20
20 156 1055 550
2
S
10.
1
2
r
44 1 2
448
11 21
122
S
Chapter 14 Sequences, Series, and the Binomial Theorem
12.
1.04r
14. Applying the Binomial Theorem to
5
2
1x
, we have
2
, 1, and 5.axb n
15.
8
2
xy
0
80 2 8
88! 8!
First Term 0 : 1
00! 8 0 !
nr r
n
rabxy x
r
0! 8!
88
xx
Cumulative Review
Cumulative Review Exercises (Chapters 1 – 14)
1.
25 32xx
2
32 2 5
32 25
xx
xx
2.
2
(5) 49
x
3.
2
6xx
Solve the related quadratic equation.
2
6
xx
The boundary points are
3 and 2.
Interval Test Value Test Conclusion
(,3)
4
2
(4) (4) 6
12 6, true
(,3) does belong to the solution set.
Chapter 14 Sequences, Series, and the Binomial Theorem
4.
63(52)4(1)
615644
xx x
xx x
5.
2
2312
33 9
xx x
23 12
33 33
xx xx
6.
3 2 4 and 4 1
3 2 3
xx
xx
7.
32 7
23 13
2 6
xyz
xyz
xyz
Multiply the second equation by 2 and add to the third equation.
Cumulative Review
We now have a system of two equations in two
variables.
55 20
520
xy
xy
Multiply the second equation by
1
and add to the
first equation.
Back-substitute 4 for x and 0 for y to find z.
32 7
xyz
8.
99
9
1
log log 8 1
log 8 1
89
xx
xx
xx
Since we cannot take a log of a negative number,
we disregard –1 and conclude that the solution set is
9.
9.
22
22
235
34 16
xy
xy
Multiply the first equation by –3 and the second
equation by 2 and solve by addition.
22
69 15
xy
2
2
235
28
x
x
10.
22
28
6
xy
xy
22
22
2
21236 8
22472 8
24 72 8
yx y
yx y
yx
Chapter 14 Sequences, Series, and the Binomial Theorem
The solution set is
14, 20 , 2, 4 .
11.
12. 35yx
14.
22
1
16 4
xy
22
2
2
52(5)(2)
(2 1)( 2) 4
(5)(2)
2524
(5)(2)
252
xx xx
xx
xx
xx
xx
xx
1
1
1
1
1
1
x
x
x
xx
xx
x
x
Cumulative Review
19. 84525720
20.
22
3
2222
333
2
55
222
xy
xy xy xy
21. 5544
5( ) 4( )
( )(5 4 )
ax ay bx by
ax y bx y
xy a b
23. 11 1
pq f
;
111
pfq
24.
22
63 14
d
Chapter 14 Sequences, Series, and the Binomial Theorem
25.
5
3
2
4
i
i
26. First, find d.
62 4
d
Next, find
30
.
a
27.
1
33
39
10 10
0.3 19
11010
110 10
a
r
28. Applying the Binomial Theorem to
4
3
2
xy
, we have
3
2 , , and 4.
axby n
4
3
234
43 2
3333
2
44 4 4 4
22 2 2
01 2 3 4
xy
xxyxy xy y
Cumulative Review
29.
2
2
() 215
fx xx
Set the denominator equal to 0 to find the domain:
30. () 2 6
fx x
;
We can not take the square root of a negative
number.
31. () ln(1 )fx x
We can only take the natural logarithm of positive
numbers.
10
x
32. Let width of the rectanglew. Then 22lw.
The perimeter of a rectangle is given by
22Pwl.
The dimension of the rectangle is 8 feet by 3 feet.
33.
1
(1 )
19610 (1 0.06)
t
AP r
P
2
11ln4
2
1
ln 4 2
k
k
k