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10.3 972 Chapter 10 Sequences, Induction, and Probability Geometric Sequences and Series H ere we are at the closing moments of a job interview. You’re shaking hands with the manager. You managed to answer all the tough questions without losing your poise, and now you’ve been offered a job. As a matter of fact, your qualifications are so terrific that you’ve been offered two jobs—one just the day before, with a rival com- pany in the same field! One company offers $30,000 the first year, with increases of 6% per year for four years after that. The other offers $32,000 the first year, with annual increases of 3% per year after that. Over a five-year period, which is the better offer? If salary raises amount to a certain percent each year, the yearly salaries over time form a geometric sequence. In this section, we investigate geometric sequences and their properties. After studying the section, you will be in a position to decide which job offer to accept: You will know which company will pay you more over five years. Geometric Sequences Figure 10.4 shows a sequence in which the number of squares is increasing. From left to right, the number of squares is 1, 5, 25, 125, and 625. In this sequence, each term after the first, 1, is obtained by multiplying the preceding term by a constant amount, namely 5. This sequence of increasing numbers of squares is an example of a geometric sequence. Objectives Find the common ratio of a geometric sequence. Write terms of a geometric sequence. Use the formula for the general term of a geometric sequence. Use the formula for the sum of the first terms of a geometric sequence. Find the value of an annuity. Use the formula for the sum of an infinite geometric series. n Section Find the common ratio of a geometric sequence. Figure 10.4 A geometric sequence of squares Definition of a Geometric Sequence A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence. The common ratio, is found by dividing any term after the first term by the term that directly precedes it. In the following examples, the common ratio is found by dividing the second term by the first term, a 2 a 1 . r,

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  • 10.3

    972 Chapter 10 Sequences, Induction, and Probability

    Geometric Sequences and Series

    Here we are at the closing moments of ajob interview. Youre shaking handswith the manager. You managed to

    answer all the tough questions withoutlosing your poise, and now youve

    been offered a job. As a matterof fact, your qualifications areso terrific that youve beenoffered two jobsone just theday before, with a rival com-pany in the same field! Onecompany offers $30,000 thefirst year, with increases of 6%per year for four years after

    that. The other offers $32,000 the first year,with annual increases of 3% per year after

    that. Over a five-year period, which is the better offer?If salary raises amount to a certain percent each year, the yearly salaries over

    time form a geometric sequence. In this section, we investigate geometric sequencesand their properties. After studying the section, you will be in a position to decidewhich job offer to accept: You will know which company will pay you more overfive years.

    Geometric SequencesFigure 10.4 shows a sequence in which the number of squares is increasing. Fromleft to right, the number of squares is 1, 5, 25, 125, and 625. In this sequence, eachterm after the first, 1, is obtained by multiplying the preceding term by a constantamount, namely 5. This sequence of increasing numbers of squares is an example ofa geometric sequence.

    Objectives

    Find the common ratio of ageometric sequence.

    Write terms of a geometricsequence.

    Use the formula for thegeneral term of a geometricsequence.

    Use the formula for the sumof the first terms of ageometric sequence.

    Find the value of an annuity. Use the formula for the sum

    of an infinite geometric series.

    n

    Sec t i on

    Find the common ratio of ageometric sequence.

    Figure 10.4 A geometric sequence of squares

    Definition of a Geometric SequenceA geometric sequence is a sequence in which each term after the first is obtainedby multiplying the preceding term by a fixed nonzero constant. The amount bywhich we multiply each time is called the common ratio of the sequence.

    The common ratio, is found by dividing any term after the first term by theterm that directly precedes it. In the following examples, the common ratio is found

    by dividing the second term by the first term,a2a1

    .

    r,

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 972

  • Section 10.3 Geometric Sequences and Series 973

    Figure 10.5 shows a partial graph of the first geometric sequence in our list.Thegraph forms a set of discrete points lying on the exponential function This illustrates that a geometric sequence with a positive common ratio other than 1is an exponential function whose domain is the set of positive integers.

    How do we write out the terms of a geometric sequence when the first termand the common ratio are known? We multiply the first term by the common ratioto get the second term, multiply the second term by the common ratio to get thethird term, and so on.

    Writing the Terms of a Geometric Sequence

    Write the first six terms of the geometric sequence with first term 6 and commonratio

    Solution The first term is 6.The second term is or 2.The third term is orThe fourth term is or and so on. The first six terms are

    Check Point 1 Write the first six terms of the geometric sequence with firstterm 12 and common ratio

    The General Term of a Geometric SequenceConsider a geometric sequence whose first term is and whose common ratio is We are looking for a formula for the general term, Lets begin by writing the firstsix terms.The first term is The second term is The third term is or The fourth term is or and so on. Starting with and multiplying eachsuccessive term by the first six terms are

    Can you see that the exponent on is 1 less than the subscript of denotingthe term number?

    a3: third term=a1r2

    One less than 3, or 2, isthe exponent on r.

    a4: fourth term=a1r3

    One less than 4, or 3, isthe exponent on r.

    ar

    a1r,

    a2, secondterm

    a1,

    a1, firstterm

    a1r2,

    a3, thirdterm

    a1r3,

    a4, fourthterm

    a1r4,

    a5, fifthterm

    a1r5.

    a6, sixthterm

    r,a1a1r

    3,a1r2 # r,

    a1r2.a1r # r,a1r.a1 .

    an .r.a1

    12 .

    6, 2, 23

    , 29

    , 227

    , and 2

    81.

    29 ,

    23# 1

    3 ,23 .

    2 # 13 ,6 # 13 ,13 .

    EXAMPLE 1

    f1x2 = 5x - 1.

    Write terms of a geometricsequence.

    Geometric sequence Common ratio

    1, 5, 25, 125, 625, r =51

    = 5

    4, 8, 16, 32, 64, r =84

    = 2

    6, 24, 96, -48,-12, r =-12

    6= -2

    9, 1, - 13

    , 19

    , -3, r =-39

    = - 13

    n

    an

    1 2 3 4 5

    255075

    100125

    Figure 10.5 The graphof 5an6 = 1, 5, 25, 125,

    Use the formula for the generalterm of a geometric sequence.

    Study TipWhen the common ratio of a geometric sequence is negative, thesigns of the terms alternate.

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 973

  • 974 Chapter 10 Sequences, Induction, and Probability

    Thus, the formula for the term is

    an=a1rn1.

    One less than n, or n 1,is the exponent on r.

    nth

    General Term of a Geometric SequenceThe term (the general term) of a geometric sequence with first term andcommon ratio is

    an = a1rn - 1.

    ra1nth

    Study TipBe careful with the order of operationswhen evaluating

    First find Then multiply theresult by a1 .

    rn - 1.

    a1rn - 1.

    Using the Formula for the General Term of a Geometric Sequence

    Find the eighth term of the geometric sequence whose first term is and whosecommon ratio is

    Solution To find the eighth term, we replace in the formula with 8, withand with

    The eighth term is 512.We can check this result by writing the first eight terms of thesequence:

    Check Point 2 Find the seventh term of the geometric sequence whose firstterm is 5 and whose common ratio is

    In Chapter 3, we studied exponential functions of the form andused an exponential function to model the growth of the U.S. population from 1970through 2007 (Example 1 on page 437). In our next example, we revisit the countryspopulation growth over a shorter period of time, 2000 through 2006. Because ageometric sequence is an exponential function whose domain is the set of positiveintegers, geometric and exponential growth mean the same thing.

    Geometric Population Growth

    The table shows the population of the United States in 2000, with estimates given bythe Census Bureau for 2001 through 2006.

    EXAMPLE 3

    f1x2 = bx

    -3.

    -4, 8, -16, 32, -64, 128, -256, 512.

    a8 = -41-228 - 1 = -41-227 = -41-1282 = 512

    an = a1rn - 1-2.r-4,

    a1na8 ,

    -2.-4

    EXAMPLE 2

    Geometric PopulationGrowth

    Economist Thomas Malthus(17661834) predicted that popula-tion would increase as a geometricsequence and food productionwould increase as an arithmeticsequence. He concluded that even-tually population would exceedfood production. If two sequences,one geometric and one arithmetic,are increasing, the geometricsequence will eventually overtakethe arithmetic sequence, regardlessof any head start that the arithmeticsequence might initially have.

    a. Show that the population is increasing geometrically.

    b. Write the general term for the geometric sequence modeling the population ofthe United States, in millions, years after 1999.

    c. Project the U.S. population, in millions, for the year 2009.

    Solution

    a. First, we use the sequence of population growth, 281.4, 284.5, 287.6, 290.8, and soon, to divide the population for each year by the population in the preceding year.

    n

    Year 2000 2001 2002 2003 2004 2005 2006

    Population (millions) 281.4 284.5 287.6 290.8 294.0 297.2 300.5

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 974

  • Section 10.3 Geometric Sequences and Series 975

    Continuing in this manner, we will keep getting approximately 1.011.This meansthat the population is increasing geometrically with The population ofthe United States in any year shown in the sequence is approximately 1.011 timesthe population the year before.

    b. The sequence of the U.S. population growth is

    Because the population is increasing geometrically, we can find the generalterm of this sequence using

    In this sequence, and [from part (a)] We substitutethese values into the formula for the general term. This gives the general termfor the geometric sequence modeling the U.S. population, in millions, yearsafter 1999.

    c. We can use the formula for the general term, in part (b) to projectthe U.S. population for the year 2009. The year 2009 is 10 years after 1999that is, Thus, We substitute 10 for in

    The model projects that the United States will have a population ofapproximately 310.5 million in the year 2009.

    Check Point 3 Write the general term for the geometric sequence

    Then use the formula for the general term to find the eighth term.

    The Sum of the First Terms of a Geometric SequenceThe sum of the first terms of a geometric sequence, denoted by and called the

    partial sum, can be found without having to add up all the terms. Recall that thefirst terms of a geometric sequence are

    We proceed as follows:

    is the sum of the first terms of the sequence.

    Multiply both sides of the equation by

    Subtract the second equationfrom the first equation.

    Factor out on the left and on the right.

    Solve for by dividing bothsides by (assuming that

    ).r Z 11 - rSn Sn =

    a111 - rn21 - r

    .

    a1Sn Sn11 - r2 = a111 - rn2

    Sn - rSn = a1 - a1rnr.

    rSn = a1r + a1r2 + a1r3 + + a1rn - 1 + a1rn

    nSn Sn = a1 + a1r + a1r2 + + a1rn - 2 + a1rn - 1

    a1 , a1r, a1r2, , a1rn - 2, a1rn - 1.

    nnth

    Snn

    n

    3, 6, 12, 24, 48, .

    a10 = 281.411.011210 - 1 = 281.411.01129 L 310.5

    an = 281.411.0112n - 1.nn = 10.2009 - 1999 = 10.

    an ,

    an = 281.411.0112n - 1

    n

    r L 1.011.a1 = 281.4

    an = a1rn - 1.

    281.4, 284.5, 287.6, 290.8, 294.0, 297.2, 300.5, .

    r L 1.011.

    284.5281.4

    L 1.011, 287.6284.5

    L 1.011, 290.8287.6

    L 1.011

    Use the formula for the sum ofthe first terms of a geometricsequence.

    n

    P-BLTZMC10_951-1036-hr 1-12-2008 14:49 Page 975

  • 976 Chapter 10 Sequences, Induction, and Probability

    To find the sum of the terms of a geometric sequence, we need to know thefirst term, the common ratio, and the number of terms, The followingexamples illustrate how to use this formula.

    Finding the Sum of the First Terms of a Geometric Sequence

    Find the sum of the first 18 terms of the geometric sequence:

    Solution To find the sum of the first 18 terms, we replace in the formulawith 18.

    We can find the common ratio by dividing the second term of by the first term.

    Now we are ready to find the sum of the first 18 terms of

    Sn =a111 - rn2

    1 - r

    2, -8, 32, -128, .

    r =a2a1

    =-82

    = -4

    2, -8, 32, -128,

    Sn=a1(1-r

    n)

    1-r

    S18=a1(1-r

    18)

    1-r

    The first term,a1, is 2.

    We must find r,the common ratio.

    nS18 ,

    2, -8, 32, -128, .

    nEXAMPLE 4

    n.r,a1 ,

    Study TipIf the common ratio is 1, the geometricsequence is

    The sum of the first terms of thissequence is

    = na1 .

    Sn =

    a1 + a1 + a1 + + a1(''''')'''''*

    There are n terms.

    na1 :n

    a1 , a1 , a1 , a1 , .

    The Sum of the First Terms of a Geometric SequenceThe sum, of the first terms of a geometric sequence is given by

    in which is the first term and is the common ratio 1r Z 12.ra1

    Sn =a111 - rn2

    1 - r,

    nSn ,

    n

    Use the formula for the sum of the first terms of a geometric sequence.

    n

    S18 =231 - 1-42184

    1 - 1-42and

    because we want the sum of the first 18 terms.n = 18a1 1the first term2 = 2, r = -4,

    Use a calculator.

    The sum of the first 18 terms is Equivalently, this number is the18th partial sum of the sequence

    Check Point 4 Find the sum of the first nine terms of the geometric sequence:

    Using to Evaluate a Summation

    Find the following sum:

    Solution Lets write out a few terms in the sum.

    a10

    i = 1 6 # 2i = 6 # 2 + 6 # 22 + 6 # 23 + + 6 # 210

    a10

    i = 1 6 # 2i.

    SnEXAMPLE 5

    2, -6, 18, -54, .

    2, -8, 32, -128, .-27,487,790,694.

    = -27,487,790,694

    We have proved the following result:

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 976

  • TechnologyTo find

    on a graphing utility, enter

    Then press ENTER .

    SUM SEQ 16 * 2x, x, 1, 10, 12.

    a10

    i = 1 6 # 2i

    Section 10.3 Geometric Sequences and Series 977

    Do you see that each term after the first is obtained by multiplying the preceding termby 2? To find the sum of the 10 terms we need to know the first term,and the common ratio, The first term is or The common ratio is 2.

    Sn =a111 - rn2

    1 - r

    12: a1 = 12.6 # 2r.a1 ,1n = 102,

    Use the formula for the sum of the firstterms of a geometric sequence.n

    S10 =1211 - 2102

    1 - 2and

    because we are adding ten terms.n = 10a1 1the first term2 = 12, r = 2,

    Use a calculator.

    Thus,

    Check Point 5 Find the following sum:

    Some of the exercises in the previous exercise set involved situations in whichsalaries increased by a fixed amount each year. A more realistic situation is one inwhich salary raises increase by a certain percent each year. Example 6 shows howsuch a situation can be modeled using a geometric sequence.

    Computing a Lifetime Salary

    A union contract specifies that each worker will receive a 5% pay increase each yearfor the next 30 years. One worker is paid $20,000 the first year. What is this personstotal lifetime salary over a 30-year period?

    Solution The salary for the first year is $20,000.With a 5% raise, the second-yearsalary is computed as follows:

    Each year, the salary is 1.05 times what it was in the previous year. Thus, the salaryfor year 3 is 1.05 times 20,000(1.05), or The salaries for the first fiveyears are given in the table.

    20,00011.0522.

    Salary for year 2 = 20,000 + 20,00010.052 = 20,00011 + 0.052 = 20,00011.052.

    EXAMPLE 6

    a8

    i = 1 2 # 3i.

    a10

    i = 1 6 # 2i = 12,276.

    = 12,276

    Yearly Salaries

    Year 1 Year 2 Year 3 Year 4 Year 5 20,000 20,000(1.05) 20,00011.0522 20,00011.0523 20,00011.0524

    The numbers in the bottom row form a geometric sequence with andTo find the total salary over 30 years, we use the formula for the sum of the

    first terms of a geometric sequence, with

    Use a calculator.

    The total salary over the 30-year period is approximately $1,328,777.

    L 1,328,777

    =20,00031 - 11.052304

    -0.05

    S30=20,000[1-(1.05)30]

    1-1.05

    Total salaryover 30 years

    Sn =a111 - rn2

    1 - r

    n = 30.nr = 1.05.

    a1 = 20,000

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 977

  • 978 Chapter 10 Sequences, Induction, and Probability

    Check Point 6 A job pays a salary of $30,000 the first year. During the next29 years, the salary increases by 6% each year. What is the total lifetime salaryover the 30-year period?

    AnnuitiesThe compound interest formula

    gives the future value, after years, when a fixed amount of money, theprincipal, is deposited in an account that pays an annual interest rate (in decimalform) compounded once a year. However, money is often invested in small amountsat periodic intervals. For example, to save for retirement, you might decide to place$1000 into an Individual Retirement Account (IRA) at the end of each year untilyou retire. An annuity is a sequence of equal payments made at equal time periods.An IRA is an example of an annuity.

    Suppose dollars is deposited into an account at the end of each year. Theaccount pays an annual interest rate, compounded annually. At the end of thefirst year, the account contains dollars. At the end of the second year, dollars isdeposited again. At the time of this deposit, the first deposit has received interestearned during the second year. The value of the annuity is the sum of all depositsmade plus all interest paid. Thus, the value of the annuity after two years is

    The value of the annuity after three years is

    The value of the annuity after years is

    This is the sum of the terms of a geometric sequence with first term and commonratio We use the formula

    to find the sum of the terms:

    This formula gives the value of an annuity after years if interest is compounded oncea year.We can adjust the formula to find the value of an annuity if equal payments aremade at the end of each of yearly compounding periods.n

    t

    St =P31 - 11 + r2t4

    1 - 11 + r2=

    P31 - 11 + r2t4-r

    =P311 + r2t - 14

    r.

    Sn =a111 - rn2

    1 - r

    1 + r.P

    P+P(1+r)+P(1+r)2+P(1+r)3+. . .+P(1+r)t1.

    First-year depositof P dollars withinterest earnedover t 1 years

    Deposit of Pdollars at end of

    year t

    t

    P + P(1+r) + P(1+r)2.

    Second-year depositof P dollars withinterest earned for

    a year

    First-year depositof P dollars withinterest earnedover two years

    Deposit of Pdollars at end of

    third year

    P+P(1+r).

    First-year depositof P dollars withinterest earned for

    a year

    Deposit of Pdollars at end of

    second year

    PPr,

    P

    rP,tA,

    A = P11 + r2t

    Find the value of an annuity.

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 978

  • Section 10.3 Geometric Sequences and Series 979

    Determining the Value of an Annuity

    At age 25, to save for retirement, you decide to deposit $200 at the end of eachmonth into an IRA that pays 7.5% compounded monthly.

    a. How much will you have from the IRA when you retire at age 65?

    b. Find the interest.

    Solution

    a. Because you are 25, the amount that you will have from the IRA when youretire at 65 is its value after 40 years.

    Use the formula for the value of an annuity.

    A =200B a1 + 0.075

    12b

    12 #40- 1R

    0.07512

    A =PB a1 + r

    nb

    nt

    - 1Rrn

    EXAMPLE 7

    Value of an Annuity: Interest Compounded Times per YearIf is the deposit made at the end of each compounding period for an annuity at

    percent annual interest compounded times per year, the value, of the annuityafter years is

    A =PB a1 + r

    nb

    nt

    - 1Rrn

    .

    tA,nr

    P

    n

    Stashing Cash andMaking Taxes LessTaxing

    The annuity involves month-end deposits of $200:The interest rate is 7.5%:

    The interest is compounded monthly: The number of years is 40: t = 40.

    n = 12.r = 0.075.P = 200.

    =200311 + 0.006252480 - 14

    0.00625

    Using parentheses keys, this can be performedin a single step on a graphing calculator.

    L200119.8989 - 12

    0.00625

    =200311.006252480 - 14

    0.00625

    Use a calculator to find

    1.00625 yx 480 = .

    11.006252480:

    After 40 years, you will have approximately $604,765 when retiring at age 65.

    b.

    L 604,765

    As you prepare for your futurecareer, retirement probably seemsvery far away. Making regulardeposits into an IRA may not befun, but there is a special incen-tive from Uncle Sam that makes itfar more appealing. TraditionalIRAs are tax-deferred savingsplans. This means that you do notpay taxes on deposits and interestuntil you begin withdrawals, typi-cally at retirement. Before then,yearly deposits count as adjust-ments to gross income and are notpart of your taxable income. Notonly do you get a tax break now,but you ultimately earn more.This is because you do not paytaxes on interest from year toyear, allowing earnings to accu-mulate until you start with-drawals. With a tax code thatencourages long-term savings,opening an IRA early in yourcareer is a smart way to gain morecontrol over how you will spend alarge part of your life.

    $604,765-$200 12 40

    =$604,765-$96,000=$508,765

    Interest=Value of the IRA-Total deposits

    $200 per month 12 monthsper year 40 years

    The interest is approximately $508,765, more than five times the amount ofyour contributions to the IRA.

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 979

  • Check Point 7 At age 30, to save for retirement, you decide to deposit $100 at theend of each month into an IRA that pays 9.5% compounded monthly.

    a. How much will you have from the IRA when you retire at age 65?

    b. Find the interest.

    Geometric SeriesAn infinite sum of the form

    with first term and common ratio is called an infinite geometric series. How canwe determine which infinite geometric series have sums and which do not? We lookat what happens to as gets larger in the formula for the sum of the first termsof this series, namely

    If is any number between and 1, that is, the term approaches 0as gets larger. For example, consider what happens to for

    Take another look at the formula for the sum of the first terms of a geometricsequence.

    Let us replace with 0 in the formula for This change gives us a formula for thesum of an infinite geometric series with a common ratio between and 1.-1

    Sn .rn

    Sn=a1(1-r

    n)

    1-rIf 1 < r < 1, rnapproaches 0 as n gets larger.

    n

    12

    12

    =

    These numbers are approaching 0 as n gets larger.

    a b1 1

    214

    =a b2 1

    218

    =a b3 1

    2116

    =a b4 1

    2132

    =a b5 1

    2164

    = .a b6

    r = 12 :rnn

    rn-1 6 r 6 1,-1r

    Sn =a111 - rn2

    1 - r.

    nnrn

    ra1

    a1 + a1r + a1r2 + a1r3 + + a1rn - 1 +

    980 Chapter 10 Sequences, Induction, and Probability

    Use the formula for the sumof an infinite geometric series.

    The Sum of an Infinite Geometric SeriesIf (equivalently, ), then the sum of the infinite geometric series

    in which is the first term and is the common ratio, is given by

    If the infinite series does not have a sum. r 1,

    S =a1

    1 - r.

    ra1

    a1 + a1r + a1r2 + a1r3 + ,

    r 6 1-1 6 r 6 1

    P-BLTZMC10_951-1036-hr 1-12-2008 14:50 Page 980

  • To use the formula for the sum of an infinite geometric series, we need toknow the first term and the common ratio. For example, consider

    With the condition that is met, so the infinite geometric series has a

    sum given by The sum of the series is found as follows:

    Thus, the sum of the infinite geometric series is 1. Notice how this is illustrated inFigure 10.6. As more terms are included, the sum is approaching the area of onecomplete circle.

    Finding the Sum of an Infinite Geometric Series

    Find the sum of the infinite geometric series:

    Solution Before finding the sum, we must find the common ratio.

    Because the condition that is met.Thus, the infinite geometric serieshas a sum.

    S =a1

    1 - r

    r 6 1r = - 12 ,

    r =a2a1

    =-

    316

    38

    = - 316

    # 83

    = - 12

    38 -

    316 +

    332 -

    364 + .

    EXAMPLE 8

    12

    +14

    +18

    +1

    16+

    132

    + =a1

    1 - r=

    12

    1 -12

    =

    1212

    = 1.

    S =a1

    1 - r.

    r 6 1r =12

    ,

    12

    14

    +18

    +116

    +132

    + +. . ..First term, a1, is .

    Common ratio, r, is .

    12

    r = = 2 =1414

    12

    12

    a2a1

    Section 10.3 Geometric Sequences and Series 981

    This is the formula for the sum of an infinite

    geometric series. Let and r = - 12

    .a1 =38

    Thus, the sum of is Put in an informal way, as we continue to add more and more terms, the sum is approximately

    Check Point 8 Find the sum of the infinite geometric series:

    We can use the formula for the sum of an infinite geometric series to express arepeating decimal as a fraction in lowest terms.

    3 + 2 + 43 +89 + .

    14 .

    14 .

    38 -

    316 +

    332 -

    364 +

    =

    38

    1 - a-12b

    =

    3832

    =38

    # 23

    =14

    qq

    qq~

    ~

    ~

    ~

    116

    116

    132

    Figure 10.6 The sumis

    approaching 1.

    12 +

    14 +

    18 +

    116 +

    132 +

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  • 982 Chapter 10 Sequences, Induction, and Probability

    Writing a Repeating Decimal as a Fraction

    Express as a fraction in lowest terms.

    Solution

    Observe that is an infinite geometric series with first term and common ratio Because the condition that is met.Thus, we can use our formula to findthe sum.Therefore,

    An equivalent fraction for is

    Check Point 9 Express as a fraction in lowest terms.Infinite geometric series have many applications, as illustrated in Example 10.

    Tax Rebates and the Multiplier Effect

    A tax rebate that returns a certain amount of money to taxpayers can have a totaleffect on the economy that is many times this amount. In economics, this phenom-enon is called the multiplier effect. Suppose, for example, that the governmentreduces taxes so that each consumer has $2000 more income. The governmentassumes that each person will spend 70% of this The individuals andbusinesses receiving this $1400 in turn spend 70% of it creating extraincome for other people to spend, and so on. Determine the total amount spent onconsumer goods from the initial $2000 tax rebate.

    Solution The total amount spent is given by the infinite geometric series

    The first term is 1400: The common ratio is 70%, or Becausethe condition that is met. Thus, we can use our formula to find the

    sum. Therefore,

    This means that the total amount spent on consumer goods from the initial $2000rebate is approximately $4667.

    Check Point 10 Rework Example 10 and determine the total amount spent onconsumer goods with a $1000 tax rebate and 80% spending down the line.

    1400 + 980 + 686 + =a1

    1 - r=

    14001 - 0.7

    L 4667.

    r 6 1r = 0.7,0.7: r = 0.7.a1 = 1400.

    1400+980+686+. . ..

    70% of1400

    70% of980

    1= $9802,1= $14002.

    EXAMPLE 10

    0.9

    79 .0.7

    0.7 =a1

    1 - r=

    710

    1 -1

    10

    =

    710910

    =7

    10# 10

    9=

    79

    .

    r 6 1r = 110 ,

    110 .

    7100.7

    0.7 = 0.7777 =7

    10+

    7100

    +7

    1000+

    710,000

    +

    0.7

    EXAMPLE 9

    $1400

    $980

    $686

    70% is spent.

    70% is spent.

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 982

  • Section 10.3 Geometric Sequences and Series 983

    Exercise Set 10.3

    Practice ExercisesIn Exercises 18, write the first five terms of each geometric sequence.

    1. 2.

    3. 4.

    5. 6.

    7. 8.

    In Exercises 916, use the formula for the general term (the nthterm) of a geometric sequence to find the indicated term of eachsequence with the given first term, and common ratio,

    9. Find when

    10. Find when

    11. Find when

    12. Find when

    13. Find when

    14. Find when

    15. Find when

    16. Find when

    In Exercises 1724,write a formula for the general term (the nth term)of each geometric sequence. Then use the formula for to find the seventh term of the sequence.

    17. 3, 12, 48, 192, 18. 3, 15, 75, 375,

    19. 18, 6, 2, 20. 12, 6, 3,

    21. 22.

    23.

    24.

    Use the formula for the sum of the first terms of a geometricsequence to solve Exercises 2530.

    25. Find the sum of the first 12 terms of the geometric sequence:

    26. Find the sum of the first 12 terms of the geometric sequence:

    27. Find the sum of the first 11 terms of the geometric sequence:

    28. Find the sum of the first 11 terms of the geometric sequence:

    29. Find the sum of the first 14 terms of the geometric sequence:

    30. Find the sum of the first 14 terms of the geometric sequence:

    In Exercises 3136, find the indicated sum. Use the formula for thesum of the first terms of a geometric sequence.

    31. 32. 33.

    34. 35. 36. a6

    i = 1 A13 B

    i + 1a

    6

    i = 1 A12 B

    i + 1a

    7

    i = 1 41-32i

    a10

    i = 1 5 # 2ia

    6

    i = 1 4ia

    8

    i = 1 3i

    n

    - 124 , 112 , -

    16 ,

    13 , .

    - 32 , 3, -6, 12, .

    4, -12, 36, -108, .

    3, -6, 12, -24, .

    3, 6, 12, 24, .

    2, 6, 18, 54, .

    n

    0.0007, -0.007, 0.07, -0.7, 0.0004, -0.004, 0.04, -0.4,

    5, -1, 15 , - 125 , .1.5, -3, 6, -12,

    32 , .

    23 , .

    a7 ,an

    a1 = 40,000, r = 0.1.a8

    a1 = 1,000,000, r = 0.1.a8

    a1 = 8000, r = - 12 .a30

    a1 = 1000, r = - 12 .a40

    a1 = 4, r = -2.a12

    a1 = 5, r = -2.a12

    a1 = 5, r = 3.a8

    a1 = 6, r = 2.a8

    r.a1 ,

    an = -6an - 1 , a1 = -2an = -5an - 1 , a1 = -6

    an = -3an - 1 , a1 = 10an = -4an - 1 , a1 = 10

    a1 = 24, r = 13a1 = 20, r =12

    a1 = 4, r = 3a1 = 5, r = 3

    In Exercises 3744, find the sum of each infinite geometric series.

    37. 38.

    39. 40.

    41. 42.

    43. 44.

    In Exercises 4550, express each repeating decimal as a fractionin lowest terms.

    45.

    46.

    47.

    48.

    49. 50.

    In Exercises 5156, the general term of a sequence is given.Determine whether the sequence is arithmetic, geometric, orneither. If the sequence is arithmetic, find the common difference;if it is geometric, find the common ratio.

    51. 52.

    53. 54.

    55. 56.

    Practice PlusIn Exercises 5762, let

    and

    57. Find 58. Find

    59. Find the difference between the sum of the first 10 terms ofand the sum of the first 10 terms of

    60. Find the difference between the sum of the first 11 terms ofand the sum of the first 11 terms of

    61. Find the product of the sum of the first 6 terms of andthe sum of the infinite series containing all the terms of

    62. Find the product of the sum of the first 9 terms of andthe sum of the infinite series containing all the terms of

    In Exercises 6364, find and for each geometric sequence.

    63. 64. 2, a2 , a3 , -548, a2 , a3 , 27

    a3a2

    5cn6.5an6

    5cn6.5an6

    5bn6.5an6

    5bn6.5an6

    a11 + b11 .a10 + b10 .

    5cn6 = -2, 1, - 12 , 14 , .

    5bn6 = 10, -5, -20, -35, ,

    5an6 = -5, 10, -20, 40, ,

    an = n2 - 3an = n2 + 5

    an = A12 Bnan = 2n

    an = n - 3an = n + 5

    0.5290.257

    0.83 =83

    100+

    8310,000

    +83

    1,000,000+

    0.47 =47

    100+

    4710,000

    +47

    1,000,000+

    0.1 =1

    10+

    1100

    +1

    1000+

    110,000

    +

    0.5 =5

    10+

    5100

    +5

    1000+

    510,000

    +

    a

    q

    i = 1 121-0.72i - 1a

    q

    i = 1 81-0.32i - 1

    3 - 1 +13

    -19

    + 1 -12

    +14

    -18

    +

    5 +56

    +562

    +563

    + 3 +34

    +342

    +343

    +

    1 +14

    +1

    16+

    164

    + 1 +13

    +19

    +1

    27+

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 983

  • 984 Chapter 10 Sequences, Induction, and Probability

    Application ExercisesUse the formula for the general term (the nth term) of a geometricsequence to solve Exercises 6568.

    In Exercises 6566, suppose you save $1 the first day of a month,$2 the second day, $4 the third day, and so on.That is, each day yousave twice as much as you did the day before.

    65. What will you put aside for savings on the fifteenth day of themonth?

    66. What will you put aside for savings on the thirtieth day of themonth?

    67. A professional baseball player signs a contract with a beginningsalary of $3,000,000 for the first year and an annual increase of4% per year beginning in the second year.That is, beginning inyear 2, the athletes salary will be 1.04 times what it was inthe previous year. What is the athletes salary for year 7 of thecontract? Round to the nearest dollar.

    68. You are offered a job that pays $30,000 for the first year withan annual increase of 5% per year beginning in the secondyear. That is, beginning in year 2, your salary will be 1.05times what it was in the previous year. What can you expectto earn in your sixth year on the job?

    In Exercises 6970, you will develop geometric sequences thatmodel the population growth for California and Texas, the twomost-populated U.S. states.

    69. The table shows population estimates for California from2003 through 2006 from the U.S. Census Bureau.

    Use the formula for the sum of the first terms of a geometricsequence to solve Exercises 7176.

    In Exercises 7172, you save $1 the first day of a month, $2 thesecond day, $4 the third day, continuing to double your savingseach day.

    71. What will your total savings be for the first 15 days?

    72. What will your total savings be for the first 30 days?

    73. A job pays a salary of $24,000 the first year. During the next19 years, the salary increases by 5% each year. What is thetotal lifetime salary over the 20-year period? Round to thenearest dollar.

    74. You are investigating two employment opportunities. CompanyA offers $30,000 the first year. During the next four years, thesalary is guaranteed to increase by 6% per year. Company Boffers $32,000 the first year, with guaranteed annual increases of3% per year after that. Which company offers the better totalsalary for a five-year contract? By how much? Round to thenearest dollar.

    75. A pendulum swings through an arc of 20 inches. On eachsuccessive swing, the length of the arc is 90% of the previouslength.

    After 10 swings, what is the total length of the distance thependulum has swung?

    76. A pendulum swings through an arc of 16 inches. On eachsuccessive swing, the length of the arc is 96% of the previouslength.

    After 10 swings, what is the total length of the distance thependulum has swung?

    Use the formula for the value of an annuity to solve Exercises7784. Round answers to the nearest dollar.

    77. To save money for a sabbatical to earn a masters degree, youdeposit $2000 at the end of each year in an annuity that pays7.5% compounded annually.

    a. How much will you have saved at the end of five years?

    b. Find the interest.

    78. To save money for a sabbatical to earn a masters degree, youdeposit $2500 at the end of each year in an annuity that pays6.25% compounded annually.

    a. How much will you have saved at the end of five years?

    b. Find the interest.

    79. At age 25, to save for retirement, you decide to deposit $50 atthe end of each month in an IRA that pays 5.5% compoundedmonthly.

    a. How much will you have from the IRA when you retireat age 65?

    b. Find the interest.

    16, 0.96(16), (0.96)2(16), (0.96)3(16), . . .

    1stswing

    2ndswing

    3rdswing

    4thswing

    20, 0.9(20), 0.92(20), 0.93(20), . . .

    1stswing

    2ndswing

    3rdswing

    4thswing

    n

    Year 2003 2004 2005 2006

    Population inmillions

    35.48 35.89 36.13 36.46

    Year 2003 2004 2005 2006

    Population inmillions

    22.12 22.49 22.86 23.41

    a. Divide the population for each year by the population inthe preceding year. Round to two decimal places andshow that California has a population increase that isapproximately geometric.

    b. Write the general term of the geometric sequencemodeling Californias population, in millions, yearsafter 2002.

    c. Use your model from part (b) to project Californiaspopulation, in millions, for the year 2010. Round to twodecimal places.

    70. The table shows population estimates for Texas from 2003through 2006 from the U.S. Census Bureau.

    n

    a. Divide the population for each year by the population inthe preceding year. Round to two decimal places andshow that Texas has a population increase that is approx-imately geometric.

    b. Write the general term of the geometric sequence modelingTexass population, in millions, years after 2002.

    c. Use your model from part (b) to project Texass popula-tion, in millions, for the year 2010. Round to two decimalplaces.

    n

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  • Section 10.3 Geometric Sequences and Series 985

    80. At age 25, to save for retirement, you decide to deposit $75 atthe end of each month in an IRA that pays 6.5% compoundedmonthly.

    a. How much will you have from the IRA when you retireat age 65?

    b. Find the interest.

    81. To offer scholarship funds to children of employees, a companyinvests $10,000 at the end of every three months in an annuitythat pays 10.5% compounded quarterly.

    a. How much will the company have in scholarship funds atthe end of ten years?

    b. Find the interest.

    82. To offer scholarship funds to children of employees, a companyinvests $15,000 at the end of every three months in an annuitythat pays 9% compounded quarterly.

    a. How much will the company have in scholarship funds atthe end of ten years?

    b. Find the interest.

    83. Here are two ways of investing $30,000 for 20 years.

    Lump-SumDeposit Rate Time

    $40,000 6.5% compoundedannually

    25 years

    PeriodicDeposits Rate Time

    $1600 at theend of eachyear

    6.5% compoundedannually

    25 years

    U

    Lump-SumDeposit Rate Time

    $30,000 5% compoundedannually

    20 years

    PeriodicDeposits Rate Time

    $1500 at theend of eachyear

    5% compoundedannually

    20 years

    After 20 years, how much more will you have from thelump-sum investment than from the annuity?

    84. Here are two ways of investing $40,000 for 25 years.

    After 25 years, how much more will you have from thelump-sum investment than from the annuity?

    Use the formula for the sum of an infinite geometric series to solveExercises 8587.

    85. A new factory in a small town has an annual payroll of $6 million.It is expected that 60% of this money will be spent in the town byfactory personnel.The people in the town who receive this money

    Function Series

    f1x2 =2B1 - a1

    3b

    xR1 -

    13

    2 + 2a13b + 2a

    13b

    2

    + 2a13b

    3

    +

    are expected to spend 60% of what they receive in the town, andso on. What is the total of all this spending, called the totaleconomic impact of the factory, on the town each year?

    86. How much additional spending will be generated by a$10 billion tax rebate if 60% of all income is spent?

    87. If the shading process shown in the figure is continued indefi-nitely, what fractional part of the largest square will eventuallybe shaded?

    Writing in Mathematics88. What is a geometric sequence? Give an example with your

    explanation.

    89. What is the common ratio in a geometric sequence?

    90. Explain how to find the general term of a geometric sequence.

    91. Explain how to find the sum of the first terms of a geometricsequence without having to add up all the terms.

    92. What is an annuity?

    93. What is the difference between a geometric sequence and aninfinite geometric series?

    94. How do you determine if an infinite geometric series has a sum?Explain how to find the sum of such an infinite geometric series.

    95. Would you rather have $10,000,000 and a brand new BMW,or 1 today, 2 tomorrow, 4 on day 3, 8 on day 4, 16 on day5, and so on, for 30 days? Explain.

    96. For the first 30 days of a flu outbreak, the number of studentson your campus who become ill is increasing. Which is worse:The number of students with the flu is increasing arithmeticallyor is increasing geometrically? Explain your answer.

    Technology Exercises97. Use the (sequence) capability of a graphing utility and

    the formula you obtained for to verify the value you foundfor in any three exercises from Exercises 1724.

    98. Use the capability of a graphing utility to calculate thesum of a sequence to verify any three of your answers toExercises 3136.

    In Exercises 99100, use a graphing utility to graph the function.Determine the horizontal asymptote for the graph of and discussits relationship to the sum of the given series.

    99.

    f

    a7

    an SEQ

    n

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  • 986 Chapter 10 Sequences, Induction, and Probability

    100. 109. In a pest-eradication program, sterilized male flies arereleased into the general population each day. Ninety percentof those flies will survive a given day. How many flies shouldbe released each day if the long-range goal of the program isto keep 20,000 sterilized flies in the population?

    110. You are now 25 years old and would like to retire at age 55with a retirement fund of $1,000,000. How much should youdeposit at the end of each month for the next 30 years in anIRA paying 10% annual interest compounded monthly toachieve your goal? Round to the nearest dollar.

    Group Exercise111. Group members serve as a financial team analyzing the three

    options given to the professional baseball player described inthe chapter opener on page 951.As a group, determine whichoption provides the most amount of money over the six-yearcontract and which provides the least. Describe one advan-tage and one disadvantage to each option.

    Preview ExercisesExercises 112114 will help you prepare for the material coveredin the next section.

    In Exercises 112113, show that

    is true for the given value of

    112. Show that

    113. Show that

    114. Simplify:k1k + 1212k + 12

    6+ 1k + 122.

    1 + 2 + 3 + 4 + 5 =515 + 12

    2.n = 5:

    1 + 2 + 3 =313 + 12

    2.n = 3:

    n.

    1 + 2 + 3 + + n =n1n + 12

    2

    Function Series

    f1x2 =431 - 10.62x4

    1 - 0.6 4 + 410.62 + 410.622 + 410.623 +

    Critical Thinking ExercisesMake Sense? In Exercises 101104, determine whethereach statement makes sense or does not make sense, and explainyour reasoning.

    101. Theres no end to the number of geometric sequences thatI can generate whose first term is 5 if I pick nonzero num-bers and multiply 5 by each value of repeatedly.

    102. Ive noticed that the big difference between arithmetic andgeometric sequences is that arithmetic sequences are based onaddition and geometric sequences are based on multiplication.

    103. I modeled Californias population growth with a geometricsequence, so my model is an exponential function whosedomain is the set of natural numbers.

    104. I used a formula to find the sum of the infinite geometricseries and then checked my answer byactually adding all the terms.

    In Exercises 105108, determine whether each statement is true orfalse. If the statement is false, make the necessary change(s) toproduce a true statement.

    105. The sequence is an example of a geometricsequence.

    106. The sum of the geometric series canonly be estimated without knowing precisely what termsoccur between and

    107.

    108. If the term of a geometric sequence is the common ratio is 12 .

    an = 310.52n - 1,nth

    10 - 5 +52

    -54

    + =10

    1 -12

    1512 .

    18

    12 +

    14 +

    18 + +

    1512

    2, 6, 24, 120,

    3 + 1 + 13 +19 +

    rr

    Mid-Chapter Check PointWhat You Know: We learned that a sequence is a functionwhose domain is the set of positive integers. In an arith-metic sequence, each term after the first differs from thepreceding term by a constant, the common difference, Ina geometric sequence, each term after the first is obtainedby multiplying the preceding term by a nonzero constant,the common ratio, We found the general term of arith-metic sequences and geometricsequences and used these formulas to findparticular terms. We determined the sum of the first

    terms of arithmetic sequences and

    geometric sequences Finally, we

    determined the sum of an infinite geometric series,

    BSn = a111 - rn2

    1 - rR .

    cSn =n

    2 1a1 + an2 d

    n3an = a1rn - 14

    3an = a1 + 1n - 12d4r.

    d. In Exercises 14, write the first five terms of each sequence.Assume that represents the common difference of an arithmeticsequence and represents the common ratio of a geometricsequence.

    1. 2.

    3. 4.

    In Exercises 57, write a formula for the general term (the term)of each sequence. Then use the formula to find the indicated term.

    5. 6.

    7.32

    , 1, 12

    , 0, ; a30

    3, 6, 12, 24, ; a102, 6, 10, 14, ; a20

    nth

    a1 = 3, an = -an - 1 + 4a1 = 5, r = -3

    a1 = 5, d = -3an = 1-12n + 1 n

    1n - 12!

    rd

    Chap te r 10a1 + a1r + a1r2 + a1r3 + , if -1 6 r 6 1S = a11 - r .

    P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 986