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13.2.11: Chapter 11

  • Page ID
    117286
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    Try It

    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.

    1. Pn = 293⋅1.026an
    2. ⓑ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.

    1. ⓐ35
    2. ⓑ330

    2.

    1. x5−5x4y+10x3y2−10x2y3+5xy4−y5
    2. ⓑ 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.

    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

    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.

    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 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.

    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 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.

    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.

    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!(nr)!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!(nr)!.

    3.

    The Binomial Theorem is defined as (x+y)n=nk=0(nk)xnkyk 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

    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

    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%


    13.2.11: Chapter 11 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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