13.2.11: Chapter 11
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11.1 Sequences and Their Notations
1.
The first five terms are {1,6, 11, 16, 21}.
2.
The first five terms are {−2, 2, −32, 1,−58}.
3.
The first six terms are {2,5,54,10,250,15}.
4.
an=(−1)n+19n
5.
an=−3n4n
6.
an=en−3
7.
{2, 5, 11, 23, 47}
8.
{0, 1, 1, 1, 2, 3, 52,176}.
9.
The first five terms are {1, 32, 4,15,72}.
11.2 Arithmetic Sequences
1.
The sequence is arithmetic. The common difference is –2.
2.
The sequence is not arithmetic because 3−1≠6−3.
3.
{1, 6, 11, 16, 21}
4.
a2=2
5.
a1=25an=an−1+12,for n≥2
6.
an=53−3n
7.
There are 11 terms in the sequence.
8.
The formula is Tn=10+4n, and it will take her 42 minutes.
11.3 Geometric Sequences
1.
The sequence is not geometric because 105≠1510 .
2.
The sequence is geometric. The common ratio is 15 .
3.
{18,6,2,23,29}
4.
a1=2an=23an−1for n≥2
5.
a6=16,384
6.
an=−(−3)n−1
7.
- ⓐ Pn = 293⋅1.026an
- ⓑThe number of hits will be about 333.
11.4 Series and Their Notations
1.
38
2.
26.4
3.
328
4.
−280
5.
$2,025
6.
≈2,000.00
7.
9,840
8.
$275,513.31
9.
The sum is not defined.
10.
The sum of the infinite series is defined.
11.
The sum of the infinite series is defined.
12.
3
13.
The series is not geometric.
14.
−311
15.
$32,775.87
11.5 Counting Principles
1.
7
2.
There are 60 possible breakfast specials.
3.
120
4.
60
5.
12
6.
P(7,7)=5,040
7.
P(7,5)=2,520
8.
C(10,3)=120
9.
64 sundaes
10.
840
11.6 Binomial Theorem
1.
- ⓐ35
- ⓑ330
2.
- ⓐ x5−5x4y+10x3y2−10x2y3+5xy4−y5
- ⓑ 8x3+60x2y+150xy2+125y3
3.
−10,206x4y5
11.7 Probability
1.
| Outcome | Probability |
|---|---|
| Heads | 12 |
| Tails | 12 |
2.
23
3.
713
4.
213
5.
56
6.
a. 191;b. 591;c. 8691
11.1 Section Exercises
1.
A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.
3.
Yes, both sets go on indefinitely, so they are both infinite sequences.
5.
A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1.
7.
First four terms: −8,−163,−4,−165
9.
First four terms: 2,12,827,14 .
11.
First four terms: 1.25,−5,20,−80 .
13.
First four terms: 13,45,97,169 .
15.
First four terms: −45,4,−20,100
17.
13,45,97,169,2511,31,44,59
19.
−0.6,−3,−15,−20,−375,−80,−9375,−320
21.
an=n2+3
23.
an=2n2nor 2n−1n
25.
an=(−12)n−1
27.
First five terms: 3,−9,27,−81,243
29.
First five terms: −1,1,−9,2711,8915
31.
124,1, 14,32,94,814,21878,531,44116
33.
2,10,12,145,45,2,10,12
35.
a1=−8,an=an−1+n
37.
a1=35,an=an−1+3
39.
720
41.
665,280
43.
First four terms: 1,12,23,32
45.
First four terms: −1,2,65,2411
47.
49.
51.
53.
an=2n−2
55.
a1=6,an=2an−1−5
57.
First five terms: 2937, 152111, 716333, 3188999, 137242997
59.
First five terms: 2, 3, 5, 17, 65537
61.
a10=7,257,600
63.
First six terms: 0.042, 0.146, 0.875, 2.385, 4.708
65.
First four terms: 5.975, 2.765, 185.743, 1057.25, 6023.521
67.
If an=−421 is a term in the sequence, then solving the equation −421=−6−8n for n will yield a non-negative integer. However, if −421=−6−8n, then n=51.875 so an=−421 is not a term in the sequence.
69.
a1=1,a2=0,an=an−1−an−2
71.
(n+2)!(n−1)!=(n+2)·(n+1)·(n)·(n−1)·...·3·2·1(n−1)·...·3·2·1=n(n+1)(n+2)=n3+3n2+2n
11.2 Section Exercises
1.
A sequence where each successive term of the sequence increases (or decreases) by a constant value.
3.
We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.
5.
Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.
7.
The common difference is 12
9.
The sequence is not arithmetic because 16−4≠64−16.
11.
0,23,43,2,83
13.
0,−5,−10,−15,−20
15.
a4=19
17.
a6=41
19.
a1=2
21.
a1=5
23.
a1=6
25.
a21=−13.5
27.
−19,−20.4,−21.8,−23.2,−24.6
29.
a1=17; an=an−1+9n≥2
31.
a1=12; an=an−1+5n≥2
33.
a1=8.9; an=an−1+1.4n≥2
35.
a1=15; an=an−1+14n≥2
37.
1=16; an=an−1−1312n≥2
39.
a1=4;an=an−1+7;a14=95
41.
First five terms: 20,16,12,8,4.
43.
an=1+2n
45.
an=−105+100n
47.
an=1.8n
49.
an=13.1+2.7n
51.
an=13n−13
53.
There are 10 terms in the sequence.
55.
There are 6 terms in the sequence.
57.
The graph does not represent an arithmetic sequence.
59.
61.
1,4,7,10,13,16,19
63.
65.
67.
Answers will vary. Examples: an=20.6n and an=2+20.4n.
69.
a11=−17a+38b
71.
The sequence begins to have negative values at the 13th term, a13=−13
73.
Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a1=3,an=an−1−3. First 4 terms: 3,0,−3,−6a31=−87
11.3 Section Exercises
1.
A sequence in which the ratio between any two consecutive terms is constant.
3.
Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.
5.
Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.
7.
The common ratio is −2
9.
The sequence is geometric. The common ratio is 2.
11.
The sequence is geometric. The common ratio is −12.
13.
The sequence is geometric. The common ratio is 5.
15.
5,1,15,125,1125
17.
800,400,200,100,50
19.
a4=−1627
21.
a7=−2729
23.
7,1.4,0.28,0.056,0.0112
25.
a=1−32,an=12an−1
27.
a1=10,an=−0.3an−1
29.
a1=35,an=16an−1
31.
a1=1512,an=−4an−1
33.
12,−6,3,−32,34
35.
an=3n−1
37.
an=0.8⋅(−5)n−1
39.
an=−(45)n−1
41.
an=3⋅(−13)n−1
43.
a12=1177,147
45.
There are 12 terms in the sequence.
47.
The graph does not represent a geometric sequence.
49.
51.
Answers will vary. Examples: a1=800,an=0.5an−1 and a1=12.5,an=4an−1
53.
a5=256b
55.
The sequence exceeds 100 at the 14th term, a14≈107.
57.
a4=−323 is the first non-integer value
59.
Answers will vary. Example: Explicit formula with a decimal common ratio: an=400⋅0.5n−1; First 4 terms: 400,200,100,50;a8=3.125
11.4 Section Exercises
1.
An nth partial sum is the sum of the first n terms of a sequence.
3.
A geometric series is the sum of the terms in a geometric sequence.
5.
An annuity is a series of regular equal payments that earn a constant compounded interest.
7.
4∑n=05n
9.
5∑k=14
11.
20∑k=18k+2
13.
S5=5(32+72)2
15.
S13=13(3.2+5.6)2
17.
7∑k=18⋅0.5k−1
19.
S5=9(1−(13)5)1−13=1219≈13.44
21.
S11=64(1−0.211)1−0.2=781,249,9849,765,625≈80
23.
The series is defined. S=21−0.8
25.
The series is defined. S=−11−(−12)
27.
29.
Sample answer: The graph of Sn seems to be approaching 1. This makes sense because ∞∑k=1(12)k is a defined infinite geometric series with S=121–(12)=1.
31.
49
33.
254
35.
S7=1472
37.
S11=552
39.
S7=5208.4
41.
S10=−1023256
43.
S=−43
45.
S=9.2
47.
$3,705.42
49.
$695,823.97
51.
ak=30−k
53.
9 terms
55.
r=45
57.
$400 per month
59.
420 feet
61.
12 feet
11.5 Section Exercises
1.
There are m+n ways for either event A or event B to occur.
3.
The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
5.
A combination; C(n,r)=n!(n−r)!r!
7.
4+2=6
9.
5+4+7=16
11.
2×6=12
13.
103=1000
15.
P(5,2)=20
17.
P(3,3)=6
19.
P(11,5)=55,440
21.
C(12,4)=495
23.
C(7,6)=7
25.
210=1024
27.
212=4096
29.
29=512
31.
8!3!=6720
33.
12!3!2!3!4!
35.
9
37.
Yes, for the trivial cases r=0 and r=1. If r=0, then C(n,r)=P(n,r)=1. If r=1, then r=1, C(n,r)=P(n,r)=n.
39.
6!2!×4!=8640
41.
6−3+8−3=8
43.
4×2×5=40
45.
4×12×3=144
47.
P(15,9)=1,816,214,400
49.
C(10,3)×C(6,5)×C(5,2)=7,200
51.
211=2048
53.
20!6!6!8!=116,396,280
11.6 Section Exercises
1.
A binomial coefficient is an alternative way of denoting the combination C(n,r). It is defined as (nr)=C(n,r)=n!r!(n−r)!.
3.
The Binomial Theorem is defined as (x+y)n=n∑k=0(nk)xn−kyk and can be used to expand any binomial.
5.
15
7.
35
9.
10
11.
12,376
13.
64a3−48a2b+12ab2−b3
15.
27a3+54a2b+36ab2+8b3
17.
1024x5+2560x4y+2560x3y2+1280x2y3+320xy4+32y5
19.
1024x5−3840x4y+5760x3y2−4320x2y3+1620xy4−243y5
21.
1x4+8x3y+24x2y2+32xy3+16y4
23.
a17+17a16b+136a15b2
25.
a15−30a14b+420a13b2
27.
3,486,784,401a20+23,245,229,340a19b+73,609,892,910a18b2
29.
x24−8x21√y+28x18y
31.
−720x2y3
33.
220,812,466,875,000y7
35.
35x3y4
37.
1,082,565a3b16
39.
1152y2x7
41.
f2(x)=x4+12x3
43.
f4(x)=x4+12x3+54x2+108x
45.
590,625x5y2
47.
k−1
49.
The expression (x3+2y2−z)5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
11.7 Section Exercises
1.
probability; The probability of an event is restricted to values between 0 and 1, inclusive of 0 and 1.
3.
An experiment is an activity with an observable result.
5.
The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A and B and a union of events A and B, the union includes either A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between 0 and 1.
7.
12.
9.
58.
11.
12.
13.
38.
15.
14.
17.
34.
19.
38.
21.
18.
23.
1516.
25.
58.
27.
113.
29.
126.
31.
1213.
33.
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | (1,1) 2 |
(1,2) 3 |
(1,3) 4 |
(1,4) 5 |
(1,5) 6 |
(1,6) 7 |
| 2 | (2,1) 3 |
(2,2) 4 |
(2,3) 5 |
(2,4) 6 |
(2,5) 7 |
(2,6) 8 |
| 3 | (3,1) 4 |
(3,2) 5 |
(3,3) 6 |
(3,4) 7 |
(3,5) 8 |
(3,6) 9 |
| 4 | (4,1) 5 |
(4,2) 6 |
(4,3) 7 |
(4,4) 8 |
(4,5) 9 |
(4,6) 10 |
| 5 | (5,1) 6 |
(5,2) 7 |
(5,3) 8 |
(5,4) 9 |
(5,5) 10 |
(5,6) 11 |
| 6 | (6,1) 7 |
(6,2) 8 |
(6,3) 9 |
(6,4) 10 |
(6,5) 11 |
(6,6) 12 |
35.
512.
37.
0.
39.
49.
41.
14.
43.
58
45.
813
47.
C(12,5)C(48,5)=12162
49.
C(12,3)C(36,2)C(48,5)=1752162
51.
C(20,3)C(60,17)C(80,20)≈12.49%
53.
C(20,5)C(60,15)C(80,20)≈23.33%
55.
20.50+23.33−12.49=31.34%
57.
C(40000000,1)C(277000000,4)C(317000000,5)=36.78%
59.
C(40000000,4)C(277000000,1)C(317000000,5)=0.11%
Review Exercises
1.
2,4,7,11
3.
13,103,1003,10003
5.
The sequence is arithmetic. The common difference is d=53.
7.
18,10,2,−6,−14
9.
a1=−20,an=an−1+10
11.
an=13n+1324
13.
r=2
15.
4, 16, 64, 256, 1024
17.
3,12,48,192,768
19.
an=−15⋅(13)n−1
21.
5∑m=0(12m+5).
23.
S11=110
25.
S9≈23.95
27.
S=1354
29.
$5,617.61
31.
6
33.
104=10,000
35.
P(18,4)=73,440
37.
C(15,6)=5005
39.
250=1.13×1015
41.
8!3!2!=3360
43.
490,314
45.
131,072a17+1,114,112a16b+4,456,448a15b2
47.
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
| 2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 |
| 3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 |
| 4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 |
| 5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 |
| 6 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 |
49.
16
51.
59
53.
49
55.
1−C(350,8)C(500,8)≈94.4%
57.
C(150,3)C(350,5)C(500,8)≈25.6%
Practice Test
1.
−14,−6,−2,0
3.
The sequence is arithmetic. The common difference is d=0.9.
5.
a1=−2,an=an−1−32;a22=−672
7.
The sequence is geometric. The common ratio is r=12.
9.
a1=1,an=−12⋅an−1
11.
15∑k=−3(3k2−56k)
13.
S7=−2604.2
15.
Total in account: $140,355.75; Interest earned: $14,355.75
17.
5×3×2×3×2=180
19.
C(15,3)=455
21.
10!2!3!2!=151,200
23.
429x1416
25.
47
27.
57
29.
C(14,3)C(26,4)C(40,7)≈29.2%
