10.3 Geometric Sequences and Series - Miami-Dade...
Transcript of 10.3 Geometric Sequences and Series - Miami-Dade...
10.3
972 Chapter 10 Sequences, Induction, and Probability
Geometric Sequences and Series
Here we are at the closing moments of ajob interview. You’re shaking hands
with the manager. You managed toanswer all the tough questions without
losing your poise, and now you’vebeen offered a job. As a matterof fact, your qualifications areso terrific that you’ve beenoffered two jobs—one 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 sumof 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,a2
a1.
r,
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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 or
The 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, Let’s 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 ,2
3# 1
3 ,23 .
2 # 13 ,6 # 1
3 ,
13 .
EXAMPLE 1
f1x2 = 5x - 1.
Write terms of a geometricsequence.
Geometric sequence Common ratio
1, 5, 25, 125, 625, Á r =
51
= 5
4, 8, 16, 32, 64, Á r =
84
= 2
6, 24, 96, Á-48,-12, r =
-126
= -2
9, 1, - 13
, 19
, Á-3, r =
-39
= - 13
n
an
1 2 3 4 5
255075
100125
Figure 10.5 The graphof 5an6 = 1, 5, 25, 125, Á
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.
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974 Chapter 10 Sequences, Induction, and Probability
Thus, the formula for the term is
an=a1rn–1.
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 country’spopulation 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(1766–1834) 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
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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 1999—that 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 - rn2
1 - r.
a1
Sn Sn11 - r2 = a111 - rn2
Sn - rSn = a1 - a1rn
r. rSn = a1r + a1r
2+ a1r
3+
Á+ a1r
n - 1+ a1r
n
nSn Sn = a1 + a1r + a1r2
+Á
+ a1rn - 2
+ a1rn - 1
a1 , a1r, a1r2, Á , a1r
n - 2, a1rn - 1.
nnth
Snn
n
3, 6, 12, 24, 48, Á .
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, Á .
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
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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, Á .
r =
a2
a1=
-82
= -4
2, -8, 32, -128, Á
Sn=a1(1-rn)
1-r
S18=a1(1-r18)
1-r
The first term,a1, is 2.
We must find r,the common ratio.
nS18 ,
2, -8, 32, -128, Á .
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 +
Á+ a1
(''''')'''''*
There are n terms.
na1 :n
a1 , a1 , a1 , a1 , Á .
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 - 1-42
and 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 Let’s write out a few terms in the sum.
a10
i = 1 6 # 2i
= 6 # 2 + 6 # 22+ 6 # 23
+Á
+ 6 # 210
a10
i = 1 6 # 2i.
SnEXAMPLE 5
2, -6, 18, -54, Á .
2, -8, 32, -128, Á .-27,487,790,694.
= -27,487,790,694
We have proved the following result:
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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 - 2
and 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 person’stotal 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 Á
20,000 20,000(1.05) 20,00011.0522 20,00011.0523 20,00011.0524 Á
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
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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)t–1.
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.
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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 +
0.07512b
12 #40
- 1R0.075
12
A =
PB a1 +
rnb
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 +
rnb
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.
L
200119.8989 - 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.
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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-rn)
1-rIf −1 < r < 1, rn
approaches 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
+Á
+ a1rn - 1
+Á
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.ƒ r ƒ Ú 1,
S =
a1
1 - r.
ra1
a1 + a1r + a1r2
+ a1r3
+Á ,
ƒ r ƒ 6 1-1 6 r 6 1
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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
ƒ r ƒ 6 1r = - 12 ,
r =
a2
a1=
-
316
38
= - 316
#83
= - 12
38 -
316 +
332 -
364 +
Á .
EXAMPLE 8
12
+
14
+
18
+
116
+
132
+Á
=
a1
1 - r=
12
1 -
12
=
1212
= 1.
S =
a1
1 - r.
ƒ r ƒ 6 1r =
12
,
12
14
+18
+116
+132
+ +. . ..First term, a1, is .
Common ratio, r, is .
12
r = ÷ = 2 =14
14
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 +
Á.
14 .
14 .3
8 -316 +
332 -
364 +
Á
=
38
1 - a-12b
=
3832
=
38
#23
=
14
qq~
~
~
~
Ω
ΩΩ
116
116
132
Figure 10.6 The sumis
approaching 1.
12 +
14 +
18 +
116 +
132 +
Á
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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 +Á
=
a1
1 - r=
14001 - 0.7
L 4667.
ƒ r ƒ 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 -
110
=
710910
=
710
#109
=
79
.
ƒ r ƒ 6 1r =110 ,
110 .7
100.7
0.7 = 0.7777 Á =
710
+
7100
+
71000
+
710,000
+Á
0.7
EXAMPLE 9
$1400
$980
$686
70% is spent.
70% is spent.
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Section 10.3 Geometric Sequences and Series 983
Exercise Set 10.3
Practice ExercisesIn Exercises 1–8, write the first five terms of each geometric sequence.
1. 2.
3. 4.
5. 6.
7. 8.
In Exercises 9–16, 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 17–24,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 25–30.
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 31–36, 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 # 2i
a6
i = 1 4i
a8
i = 1 3i
n
- 124 , 1
12 , - 16 , 13 , Á .
- 32 , 3, -6, 12, Á .
4, -12, 36, -108, Á .
3, -6, 12, -24, Á .
3, 6, 12, 24, Á .
2, 6, 18, 54, Á .
n
0.0007, -0.007, 0.07, -0.7, Á
0.0004, -0.004, 0.04, -0.4, Á
5, -1, 15 , - 125 , Á .1.5, -3, 6, -12, Á
32 , Á .2
3 , Á .
ÁÁ
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 37–44, find the sum of each infinite geometric series.
37. 38.
39. 40.
41. 42.
43. 44.
In Exercises 45–50, express each repeating decimal as a fractionin lowest terms.
45.
46.
47.
48.
49. 50.
In Exercises 51–56, 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 57–62, 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 63–64, 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 , Á .
5bn6 = 10, -5, -20, -35, Á ,
5an6 = -5, 10, -20, 40, Á ,
an = n2- 3an = n2
+ 5
an = A12 Bnan = 2n
an = n - 3an = n + 5
0.5290.257
0.83 =
83100
+
8310,000
+
831,000,000
+Á
0.47 =
47100
+
4710,000
+
471,000,000
+Á
0.1 =
110
+
1100
+
11000
+
110,000
+Á
0.5 =
510
+
5100
+
51000
+
510,000
+Á
a
q
i = 1 121-0.72i - 1
a
q
i = 1 81-0.32i - 1
3 - 1 +
13
-
19
+Á1 -
12
+
14
-
18
+Á
5 +
56
+
562 +
563 +
Á3 +
34
+
342 +
343 +
Á
1 +
14
+
116
+
164
+Á1 +
13
+
19
+
127
+Á
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 65–68.
In Exercises 65–66, 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 athlete’s salary will be 1.04 times what it was inthe previous year. What is the athlete’s 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 69–70, 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 71–76.
In Exercises 71–72, 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 Exercises77–84. Round answers to the nearest dollar.
77. To save money for a sabbatical to earn a master’s 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 master’s 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 California’s population, in millions, yearsafter 2002.
c. Use your model from part (b) to project California’spopulation, 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 modelingTexas’s population, in millions, years after 2002.
c. Use your model from part (b) to project Texas’s popula-tion, in millions, for the year 2010. Round to two decimalplaces.
n
P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 984
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 85–87.
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 - a13b
xR1 -
13
2 + 2a13b + 2a
13b
2
+ 2a13b
3
+Á
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 17–24.
98. Use the capability of a graphing utility to calculate thesum of a sequence to verify any three of your answers toExercises 31–36.
In Exercises 99–100, 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
P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 985
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 112–114 will help you prepare for the material coveredin the next section.
In Exercises 112–113, 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 +Á
+ n =
n1n + 12
2
Function Series
f1x2 =
431 - 10.62x4
1 - 0.6 4 + 410.62 + 410.622 + 410.623 +
Á
Critical Thinking ExercisesMake Sense? In Exercises 101–104, determine whethereach statement makes sense or does not make sense, and explainyour reasoning.
101. There’s 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. I’ve 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 California’s 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 105–108, 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
+Á
=
10
1 -
12
1512 .1
8
12 +
14 +
18 +
Á+
1512
2, 6, 24, 120, Á
3 + 1 +13 +
19 +
Á
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 = a1r
n - 14
3an = a1 + 1n - 12d4r.
d. In Exercises 1–4, 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 5–7, 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, Á ; a30
3, 6, 12, 24, Á ; a102, 6, 10, 14, Á ; 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 + a1r
2+ a1r
3+
Á, if -1 6 r 6 1¢S =
a1
1 - r≤ .
P-BLTZMC10_951-1036-hr 26-11-2008 16:23 Page 986