13.2.11: Chapter 11

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Try It

11.1 Sequences and Their Notations

The first five terms are {1,6, 11, 16, 21}.

The first five terms are {−2, 2, −32, 1,−58}.

The first six terms are {2,5,54,10,250,15}.

an=(−1)n+19n

an=−3n4n

an=en−3

{2, 5, 11, 23, 47}

{0, 1, 1, 1, 2, 3, 52,176}.

The first five terms are {1, 32, 4,15,72}.

11.2 Arithmetic Sequences

The sequence is arithmetic. The common difference is –2.

The sequence is not arithmetic because 3−1≠6−3.

{1, 6, 11, 16, 21}

a2=2

a1=25an=an−1+12,for n≥2

an=53−3n

There are 11 terms in the sequence.

The formula is Tn=10+4n, and it will take her 42 minutes.

11.3 Geometric Sequences

The sequence is not geometric because 105≠1510 .

The sequence is geometric. The common ratio is 15 .

{18,6,2,23,29}

a1=2an=23an−1for n≥2

a6=16,384

an=−(−3)n−1

ⓐ Pn = 293⋅1.026an
ⓑThe number of hits will be about 333.

11.4 Series and Their Notations

26.4

328

−280

$2,025

≈2,000.00

9,840

$275,513.31

The sum is not defined.

10.

The sum of the infinite series is defined.

11.

The sum of the infinite series is defined.

12.

13.

The series is not geometric.

14.

−311

15.

$32,775.87

11.5 Counting Principles

There are 60 possible breakfast specials.

120

P(7,7)=5,040

P(7,5)=2,520

C(10,3)=120

64 sundaes

10.

840

11.6 Binomial Theorem

ⓐ35
ⓑ330

ⓐ x5−5x4y+10x3y2−10x2y3+5xy4−y5
ⓑ 8x3+60x2y+150xy2+125y3

−10,206x4y5

11.7 Probability

Outcome	Probability
Heads	12
Tails	12

713

213

a. 191;b. 591;c. 8691

11.1 Section Exercises

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.

Yes, both sets go on indefinitely, so they are both infinite sequences.

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.

First four terms: −8,−163,−4,−165

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.

$Graph of a scattered plot with points at (1, 0), (2, 5/2), (3, 8/3), (4, 17/4), and (5, 24/5). The x-axis is labeled n and the y-axis is labeled a_n.$

49.

$Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.$

51.

$Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.$

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

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

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.

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.

The common difference is 12

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.

$Graph of a scattered plot with labeled points: (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31). The x-axis is labeled n and the y-axis is labeled a_n.$

61.

1,4,7,10,13,16,19

63.

$Graph of a scattered plot with labeled points: (1, 1), (2, 4), (3, 7), (4, 10), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n.$

65.

$Graph of a scattered plot with labeled points: (1, 5.5), (2, 6), (3, 6.5), (4, 7), and (5, 7.5). The x-axis is labeled n and the y-axis is labeled a_n.$

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 13^th 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

A sequence in which the ratio between any two consecutive terms is constant.

Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.

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.

The common ratio is −2

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.

$Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.$

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 14^th 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

An nth partial sum is the sum of the first n terms of a sequence.

A geometric series is the sum of the terms in a geometric sequence.

An annuity is a series of regular equal payments that earn a constant compounded interest.

4∑n=05n

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.

$Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.$

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.

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

There are m+n ways for either event A or event B to occur.

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.

A combination; C(n,r)=n!(n−r)!r!

4+2=6

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.

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

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)!.

The Binomial Theorem is defined as (x+y)n=n∑k=0(nk)xn−kyk and can be used to expand any binomial.

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

$Graph of the function f_2.$ 43.

f4(x)=x4+12x3+54x2+108x

$Graph of the function f_4.$ 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

probability; The probability of an event is restricted to values between 0 and 1, inclusive of 0 and 1.

An experiment is an activity with an observable result.

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.

12.

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