Section 14.3 Geometric Sequences and Series
93.
2
1
22
1
a
a
94.
2
2
1
35 5
35
a
a
14.3 Check Points
1.
1
1
12, 2
ar
1
2
2
3
1
12 6
2
112
12 3
24
a
a
24 8
2.
1
1
1
5, 3
n
n
ar
aar
The eighth term is 384.
6
1
8
8
1
61 3 6( 6560) 19,680
13 2
n
Sr
S
Thus,
8
1
23 19,680.
i
i
6.
1
1
30,000, 1.06
(1 )
n
n
ar
ar
Sr
Chapter 14 Sequences, Series, and the Binomial Theorem
7. a.
11
nt
r
n
r
P
A
b.
Interest Value of IRA Total deposits
$333,946 $100 12 35
$333,946 $42,000
$291,946
8.
48
32 39
The sum of this infinite geometric series is 9.
9.
99 9
0.9 0.9999 10 100 1000
10.
1000(0.8) 800, 0.8
ar
14.3 Concept and Vocabulary Check
1. geometric; common ratio
3.
1
(1 )
1
n
ar
r
; first term; common ratio
8. geometric
9. geometric
10. arithmetic
3.
2
1
30 2
15
a
ra
6.
2
8812
34
a
ra
8.
2
1
0.7 0.1
7
a
ra
9. The first term is 2.
Section 14.3 Geometric Sequences and Series
10. The first term is 2.
The second term is
24 8
.
11. The first term is 20.
The second term is
1
20 10
2
.
12. The first term is 24.
The second term is
1
24 8
3
.
13. The first term is –4.
The second term is
410 40
.
14. The first term is –3.
The second term is
310 30
.
15. The first term is
1
4
.
16. The first term is
1
.
The third term is
4
.
The fourth term is
14 4
.
The fifth term is
44 16
.
19.
12
52 52
52048 10,240
a
22.
61 5
6
11
8000 8000 250
a
1,000,000 0.0000001 0.1
81 7
1
71 6
7
34
34 34
n
aar
a
Chapter 14 Sequences, Series, and the Binomial Theorem
27.
2
1
61
18 3
a
ra
28.
2
1
61
12 2
a
ra
29.
2
32
1.5
a
ra
30.
2
1
11
55
a
ra
31.
2
1
0.004 10
0.0004
a
ra
32.
2
1
0.007 10
0.0007
a
ra
12
12
21 3 2 1 531,441
13 2
S
1
3
12
1
1312
n
ar
35.
2
1
3
ra
1
4
11
41 3
1
n
ar
Section 14.3 Geometric Sequences and Series
37.
2
1
33
3
32
2
a
ra
38.
2
1
1
124
12 2
112 1
24
a
ra
39.
8
8
1
31 3 3 1 6561
313 2
36560 19, 680 9840
22
i
i
42.
7
7
1
12 1 3
43 13
i
i
464 2 4
2
64
1
63
128
6
11 11
11
2187
45.
2
1
1
1
3
13
a
ra
11 233
a
Chapter 14 Sequences, Series, and the Binomial Theorem
47.
2
1
3
3311
43
34 434
a
ra
48.
2
1
5
5511
65
56 656
a
ra
49.
2
1
1
1
2
12
a
ra
50.
2
11
33
a
ra
51.
0.3
r
11 0
1
26 0.3 26 0.3
26 1 26
a
10.71.7
53.
1
55
59
10 10
0.5 19
11010
a
r
11099
110 10
r
47 47
100 100 100 99 99
a
56.
1
83 83
83 100 83
100 100
0.83 199
1 100 99 99
a
r
257 999 257 1000
1000 1000 1000 999
58.
1
1000 1000
0.529 1999
111000 1000
529 1000 529
1000 999 999
a
r
Section 14.3 Geometric Sequences and Series
63. The sequence is neither arithmetic nor geometric.
64. The sequence is neither arithmetic nor geometric.
66. First find
11
a and
11
b:
1
11 1
n
aar
67. From Exercise 65,
10
2560a and
10
125b .
For
{}
n
a,
10 2
5
r
68. For {}
n
a,
10 2
r
5.5( 130) 715
So,
11 11
3415 ( 715) 2700
ab
3
For
{}
n
c,
1
13
22
224
13
1
a
Sr
Chapter 14 Sequences, Series, and the Binomial Theorem
71. It is given that
4
27a. Using the formula
1
1
n
n
aar
when 4n we have:
41
3
27 8
27
r
r
72. It is given that
454a
. Using the formula
1
1
n
n
aar
when
4n
we have:
41
3
54 2
r
73. Find the total value of the lump-sum investment.
20
1 30,000 1 0.05 79,599
t
AP r
Find the total value of the annuity.
Section 14.3 Geometric Sequences and Series
74. Find the total value of the lump-sum investment.
25
1 40,000 1 0.065 193,108
t
AP r
75.
2
1
15 1 14
15
22
1
12 2 16,384
a
ra
a
On the fifteenth day, you will put aside $16,384 for savings.
77.
71
7
1.04
3,000,000 1.04
r
a
78.
1.05
r
Chapter 14 Sequences, Series, and the Binomial Theorem
79. a.
2000 to 2001
34.21 1.01
33.87
r
2001 to 2002
34.55 1.01
34.21
r
37.09
r
is approximately 1.01 for all but one division.
b.
1
1
1
33.87 1.01
n
n
n
n
aar
a
c. Since year 2020 is the 21th term, find
21
.
a
80. a.
2000 to 2001
21.27 1.02
20.85
r
2001 to 2002
21.70 1.02
21.27
r
24.92
r
is approximately 1.02 for all but one division.
b.
1
1
1
20.85 1.02
n
n
n
n
aar
a
c. Since year 2020 is the 21th term, find
21
.
a
15
1
15
11 2
1
112
1 32,768 32, 767 32, 767
11
n
ar
Sr
Your savings will be $32,767 over the 15 days.
Section 14.3 Geometric Sequences and Series
82.
2
1
22
1
a
ra
83.
1.05
r
20
1
20
24,000 1 1.05
1
n
ar
Sr
84. Company A:
1.06
r
5
1
5
30, 000 1 1.06
1
1 1 1.06
30,000 1 1.338226
0.06
30,000 0.338226 169,112.79
0.06
n
ar
Sr
Company B:
1.03
r
85. Option #1 (starting at $1,700,000 with $70,000
annual increases):
6
2
61,700,000 2,050,000
2
3(3,750,000) 11, 250,000
S
0.02
1,700,000 0.126162 10,723,806
0.02
Option #3 (starting at $1,500,000 with 9% annual
increases):
1.09
r
6
1
6
1,500, 000 1 1.09
1
111.09
1,500, 000 1 1.677100
11.09
1,500, 000 0.677100 11, 285,002
n
ar
Sr
Chapter 14 Sequences, Series, and the Binomial Theorem
87.
0.9
r
88.
0.96
r
16 0.335167364 134.067
0.04
After 10 swings, the pendulum covers a distance of approximately 134.07 inches.
90. a.
5
0.0625
1 1 2500 1 1
1
$14,163
0.0625
1
nt
r
Pn
Ar
n
b. $14,163 5 $2500 $1663
91. a.
12 40
0.055
12
0.055
12
11
50 1 1
$87,052
nt
r
Pn
Ar
Section 14.3 Geometric Sequences and Series
94. a.
410
0.09
4
0.09
4
1 1 15,000 1 1
$956,793
nt
r
n
r
n
P
A
b. $956,793 $15, 000 4 10 $356,793
96.
60% 0.6
r
1
10 .6 6
a
97.
1
4
r
Chapter 14 Sequences, Series, and the Binomial Theorem
109.
1
21 3
1
13
x
fx
110.
41 0.6
10.6
x
fx
111. makes sense
112. makes sense
113. makes sense
Section 14.3 Geometric Sequences and Series
119.
1
1
20,000 10.9
a
Sr
x
120.
11
nt
r
Pn
Ar
n
You should deposit approximately $442 per month.
121.
28 37 63 47 37 97
27 37 37
27
Chapter 14 Sequences, Series, and the Binomial Theorem
122.
2
2
24
240
xx
xx
2 1 4abc
Solve using the quadratic formula.
123.
6635
35 3535
124. The exponents begin with the exponent on
ab
and decrease by 1 in each successive term.
Mid-Chapter Check Point – Chapter 14
1.
1
(1) (1)!
n
n
n
an
11 2
1
21 3
2
11
(1) (1) 11 1
(1 1) ! 0 !
22
(1) (1) (1)(2) 2
a
a
+
+
=− =− = ⋅ =
−
=− =− =− =−
2. Using
1
(1)
n
aand
;
1
2
5
5(21)(3) 51(3) 53 2
a
a
41 3
4
51 4
5
5( 3) 5( 3) 5( 27) 135
5( 3) 5( 3) 5(81) 405
a
a
20
4(20) 2 78
a
6.
21
31
122
da a
1
30
31
(1) (1)
22
31 1 1 2
22 2 2
1(30) 2 15 2 13
2
n
aand n
nn
a
7. First find r;
Mid-Chapter Check Point
8. First find
10
a
;
21
0(2) 2
da a
9. First find r;
2
1
40 2
20
a
ra
10. First find
100
a
;
21
100 1
24 6
(1) 4(1001)(6)
da a
aand
11.
4
1
( 4)( 1) (1 4)(1 1) (2 4)(2 1) (3 4)(3 1) (4 4)(4 1)
5(0) 6(1) 7(2) 8(3) 0 6 14 24 44
i
ii
The sum of this arithmetic sequence is given by
1
()
2
nn
n
Saa
;
50
50 (1 148) 25(149) 3725
2
S
13.
6
1
3
2
i
i
Chapter 14 Sequences, Series, and the Binomial Theorem
14.
1
2
5
i
15.
1
45 45
100 100
0.45 199
11100 100
45 99 45 100 45 5
100 100 100 99 99 11
a
r
17. The arithmetic sequence is 16, 48, 80, 112, ….
First find
15
a
where
21
48 16 32da a
.
18.
8
(1 )
500,000(1 0.10)
1,071,794
t
AP r
Section 14.4 The Binomial Theorem
14.4 Check Points
1. a.
66! 6! 5 4 20
33!(6 3)! 3!3! 1
2.
4432 432
44444
(1) 4 6 41
01234
xxxxxxxxx
4.
9
(2 )xy
14.4 Concept and Vocabulary Check
1. binomial
2.
8!
2!6!
3.
!
!( )!
m
rn r
Chapter 14 Sequences, Series, and the Binomial Theorem
14.4 Exercise Set
1.
88! 8! 8 7 6
33! 8 3 ! 3!5!
5!
3215! 56
4.
11 11! 11 10!
11! 11 1 !
110!11
5.
66!
6
6!
11
1
0! 1
66!
9. Applying the Binomial Theorem to
3
2x
, we have
, 2, and 3.axb n