section 11.3 geometric sequences and series · 9, -3, 1 , - 1 3, 1 9. r =-3 9 = - 1 3 figure 11.6...

Section 11.3 Geometric Sequences and Series 1039

Preview Exercises Exercises 85–87 will help you prepare for the material covered in the next section.

85. Consider the sequence 1, -2, 4, -8, 16, . . . . Find a2

a1,

a3

a2,

a4

a3,

and a5

a4. What do you observe?

86. Consider the sequence whose nth term is an = 3 # 5n. Find a2

a1,

a3

a2,

a4

a3, and

a5

a4. What do you observe?

87. Use the formula an = a13n-1 to fi nd the seventh term of the

sequence 11, 33, 99, 297, . . . .

83. A degree-day is a unit used to measure the fuel requirements of buildings. By defi nition, each degree that the average daily temperature is below 65°F is 1 degree-day. For example, an average daily temperature of 42°F constitutes 23 degree-days. If the average temperature on January 1 was 42°F and fell 2°F for each subsequent day up to and including January 10, how many degree-days are included from January 1 to January 10?

84. Show that the sum of the fi rst n positive odd integers,

1 + 3 + 5 + g+ (2n - 1),

is n2.

that. The other offers $32,000 the fi rst year, with annual increases of 3% per year after that. Over a fi ve-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 fi ve years.

Geometric Sequences Figure 11.5 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 fi rst, 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 fi rst n terms of a geometric sequence.

� Find the value of an annuity.

� Use the formula for the sum of an infi nite geometric series.

Geometric Sequences and Series SECTION 11.3

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 qualifi cations are so terrifi c that you’ve been offered two jobs—one just the day before, with a rival company in the same fi eld! One company offers $30,000 the fi rst year, with increases of 6% per year for four years after

� Find the common ratio of a geometric sequence.

FIGURE 11.5 A geometric sequence of squares

M18_BLIT7240_06_SE_11-hr.indd 1039 13/10/12 11:22 AM

1040 Chapter 11 Sequences, Induction, and Probability

The common ratio, r, is found by dividing any term after the fi rst term by the term that directly precedes it. In the following examples, the common ratio is found by dividing the second term by the fi rst term,

a2

a1.

Geometric sequence Common ratio

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

4, 8, 16, 32, 64, . . . r =84= 2

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

6= -2

9, -3, 1, - 13

, 19

, . . . r =-39

= - 13

Figure 11.6 shows a partial graph of the fi rst geometric sequence in our list. The graph forms a set of discrete points lying on the exponential function f(x) = 5x-1. This illustrates that a geometric sequence with a positive common ratio other than 1 is an exponential function whose domain is the set of positive integers .

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

EXAMPLE 1 Writing the Terms of a Geometric Sequence

Write the fi rst six terms of the geometric sequence with fi rst term 6 and common ratio 13.

SOLUTION The fi rst term is 6. The second term is 6 # 1

3, or 2. The third term is 2 # 13, or 23. The

fourth term is 23 # 13, or 29, and so on. The fi rst six terms are

6, 2, 23

, 29

, 227

, and 281

. ● ● ●

Check Point 1 Write the fi rst six terms of the geometric sequence with fi rst term 12 and common ratio 12.

The General Term of a Geometric Sequence Consider a geometric sequence whose fi rst term is a1 and whose common ratio is r. We are looking for a formula for the general term, an. Let’s begin by writing the fi rst six terms. The fi rst term is a1. The second term is a1r. The third term is a1r # r, or a1r

2. The fourth term is a1r

2 # r, or a1r3, and so on. Starting with a1 and multiplying each

successive term by r, the fi rst six terms are

a1r,

a2, secondterm

a1,

a1, firstterm

a1r2,

a3, thirdterm

a1r3,

a4, fourthterm

a1r4,

a5, fifthterm

a1r5.

a6, sixthterm

Defi nition of a Geometric Sequence

A geometric sequence is a sequence in which each term after the fi rst is obtained by multiplying the preceding term by a fi xed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

GREAT QUESTION! What happens to the terms of a geometric sequence when the common ratio is negative?

When the common ratio is negative, the signs of the terms alternate.

n

an

1 2 3 4 5

255075

100125

FIGURE 11.6 The graph of {an} = 1, 5, 25, 125, . . .

� Write terms of a geometric sequence.

� Use the formula for the general term of a geometric sequence.



Can you see that the exponent on r is 1 less than the subscript of a denoting the 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.

Thus, the formula for the nth term is

an=a1rn–1.

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

General Term of a Geometric Sequence

The nth term (the general term) of a geometric sequence with fi rst term a1 and common ratio r is

an = a1rn-1.

GREAT QUESTION! When using a1r

n-1 to fi nd the n th term of a geometric sequence, what should I do fi rst?

Be careful with the order of operations when evaluating

a1rn-1.

First subtract 1 in the exponent and then raise r to that power. Finally, multiply the result by a1.

EXAMPLE 2 Using the Formula for the General Term of a Geometric Sequence

Find the eighth term of the geometric sequence whose fi rst term is -4 and whose common ratio is -2.

SOLUTION To fi nd the eighth term, a8, we replace n in the formula with 8, a1 with -4, and r with -2.

an = a1rn-1

a8 = -4(-2)8-1 = -4(-2)7 = -4(-128) = 512

The eighth term is 512. We can check this result by writing the fi rst eight terms of the sequence:

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

Check Point 2 Find the seventh term of the geometric sequence whose fi rst term is 5 and whose common ratio is -3.

In Chapter 4 , we studied exponential functions of the form f(x) = bx and used an exponential function to model the growth of the U.S. population from 1970 through 2010 (Example 1 on page 495 ). From 2000 through 2010, the nation’s population grew at the slowest rate since the 1930s. Consequently, in our next example, we consider the country’s population growth from 2000 through 2010. Because a geometric sequence is an exponential function whose domain is the set of positive integers, geometric and exponential growth mean the same thing.

Blitzer Bonus Geometric Population

Growth

Economist Thomas Malthus(1766–1834) predicted that population would increase as a geometric sequence and food production would increase as an arithmetic sequence. He concluded that eventually population would exceed food production. If two sequences, one geometric and one arithmetic, are increasing, the geometric sequence will eventually overtake the arithmetic sequence, regardless of any head start that the arithmetic sequence might initially have.



Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Population (millions) 281.4 284.0 286.6 289.3 292.0 294.7 297.4 300.2 303.0 305.8 308.7

EXAMPLE 3 Geometric Population Growth

The table shows the population of the United States in 2000 and 2010, with estimates given by the Census Bureau for 2001 through 2009.

a. Show that the population is increasing geometrically. b. Write the general term for the geometric sequence modeling the population

of the United States, in millions, n years after 1999. c. Project the U.S. population, in millions, for the year 2020.

SOLUTION a. First, we use the sequence of population growth, 281.4, 284.0, 286.6, 289.3,

and so on, to divide the population for each year by the population in the preceding year.

284.0281.4

� 1.009, 286.6284.0

� 1.009, 289.3286.6

� 1.009

Continuing in this manner, we will keep getting approximately 1.009. This means that the population is increasing geometrically with r � 1.009. The population of the United States in any year shown in the sequence is approximately 1.009 times the population the year before.

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

281.4, 284.0, 286.6, 289.3, 292.0, 294.7, . . . .

Because the population is increasing geometrically, we can fi nd the general term of this sequence using

an = a1rn-1.

In this sequence, a1 = 281.4 and [from part (a)] r � 1.009. We substitute these values into the formula for the general term. This gives the general term for the geometric sequence modeling the U.S. population, in millions, n years after 1999.

an = 281.4(1.009)n-1

c. We can use the formula for the general term, an, in part (b) to project the U.S. population for the year 2020. The year 2020 is 21 years after 1999—that is, 2020 - 1999 = 21. Thus, n = 21. We substitute 21 for n in an = 281.4(1.009)n-1.

a21 = 281.4(1.009)21-1 = 281.4(1.009)20 � 336.6

The model projects that the United States will have a population of approximately 336.6 million in the year 2020. ● ● ●

Check Point 3 Write the general term for the geometric sequence

3, 6, 12, 24, 48, . . . .

Then use the formula for the general term to fi nd the eighth term.



The Sum of the First n Terms of a Geometric Sequence The sum of the fi rst n terms of a geometric sequence, denoted by Sn and called the nth partial sum , can be found without having to add up all the terms. Recall that the fi rst n terms of a geometric sequence are

a1, a1r, a1r2, . . . , a1r

n-2, a1rn-1.

We proceed as follows:

Sn = a1 + a1r + a1r2 + g + a1r

n-2 + a1rn-1 Sn is the sum of the fi rst

n terms of the sequence.

rSn = a1r + a1r2 + a1r

3 + g + a1rn-1 + a1r

n Multiply both sides of the equation by r.

Sn - rSn = a1 - a1rn Subtract the second equation

from the fi rst equation.

Sn(1 - r) = a1(1 - rn) Factor out Sn on the left and a1 on the right.

Sn =a1(1 - rn)

1 - r.

Solve for Sn by dividing both sides by 1 - r (assuming that r � 1).

We have proved the following result:

Blitzer Bonus ❘ ❘ Ponzi Schemes and Geometric Sequences A ponzi scheme is an investment fraud that pays returns to existing investors from funds contributed by new investors rather than from legitimate investment activity. Here’s a simplifi ed example:

a2 Round 2

,a1 Round 1

, a3 Round 3

, a4 Round 4

, Á

The schemer takes$100 each from 4 new investors,pocketing $200and paying the

first two investors$100 each.

The schemertakes $100

each from thefirst twoinvestors.

The schemer takes$100 each from 8 new investors,pocketing $200and paying the

previous 6 investors$100 each.

The schemer takes$100 each from16 new investors,pocketing $200and paying the

previous 14 investors$100 each.

The number of investors needed to continue this Ponzi scheme,

2, 4, 8, 16, . . . ,

and the money collected in each round,

$200, $400, $800, $1600, . . . ,

form rapidly growing geometric sequences. With no legitimate earnings, the scheme requires a consistent geometric fl ow of money from new investors to continue. Ponzi schemes tend to collapse when it becomes diffi cult to recruit new investors or when a large number of investors ask to cash out.

� Use the formula for the sum of the fi rst n terms of a geometric sequence.

The Sum of the First n Terms of a Geometric Sequence

The sum, Sn, of the fi rst n terms of a geometric sequence is given by

Sn =a1(1 - rn)

1 - r,

in which a1 is the fi rst term and r is the common ratio (r � 1).

GREAT QUESTION! What is the sum of the fi rst n terms of a geometric sequence if the common ratio is 1?

If the common ratio is 1, the geometric sequence is

a1, a1, a1, a1, . . . .

The sum of the fi rst n terms of this sequence is na1 :

Sn=a1+a1+a1+Á+a1.

=na1.

There are n terms



To fi nd the sum of the terms of a geometric sequence using Sn =a1(1 - rn)

1 - r, we

need to know the fi rst term, a1, the common ratio, r, and the number of terms, n. The following examples illustrate how to use this formula.

EXAMPLE 4 Finding the Sum of the First n Terms of a Geometric Sequence

Find the sum of the fi rst 18 terms of the geometric sequence: 2, -8, 32, -128, . . . .

SOLUTION To fi nd the sum of the fi rst 18 terms, S18, we replace n in the formula with 18.

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.

We can fi nd the common ratio by dividing the second term of 2, -8, 32, -128, . . . by the fi rst term.

r =a2

a1=

-82

= -4

Now we are ready to fi nd the sum of the fi rst 18 terms of 2, -8, 32, -128, . . . .

Sn =a1(1 - rn)

1 - r

Use the formula for the sum of the fi rst n terms of a geometric sequence.

S18 =2[1 - (-4)18]

1 - (-4)

a1 (the first term) = 2, r = -4, and n = 18 because we want the sum of the fi rst 18 terms.

= -27,487,790,694 Use a calculator.

The sum of the fi rst 18 terms is -27,487,790,694. Equivalently, this number is the 18th partial sum of the sequence 2, -8, 32, -128, . . . . ● ● ●

Check Point 4 Find the sum of the fi rst nine terms of the geometric sequence: 2, -6, 18, -54, . . . .

EXAMPLE 5 Using Sn to Evaluate a Summation

Find the following sum: a10

i=1 6 # 2i.

SOLUTION Let’s write out a few terms in the sum.

a10

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



Sn =a1(1 - rn)

1 - r

Use the formula for the sum of the fi rst n terms of a geometric sequence.

S10 =12(1 - 210)

1 - 2

a1 (the first term) = 12, r = 2, and n = 10 because we are adding 10 terms.

= 12,276 Use a calculator.

Do you see that each term after the fi rst is obtained by multiplying the preceding term by 2? To fi nd the sum of the 10 terms (n = 10), we need to know the fi rst term, a1, and the common ratio, r. The fi rst term is 6 # 2 or 12: a1 = 12. The common ratio is 2.

Thus,

a10

i=1 6 # 2i = 12,276. ● ● ●

Check Point 5 Find the following sum: a8

i=1 2 # 3i.

Some of the exercises in the previous Exercise Set involved situations in which salaries increased by a fi xed amount each year. A more realistic situation is one in which salary raises increase by a certain percent each year. Example 6 shows how such a situation can be modeled using a geometric sequence.

EXAMPLE 6 Computing a Lifetime Salary

A union contract specifi es that each worker will receive a 5% pay increase each year for the next 30 years. One worker is paid $20,000 the fi rst year. What is this person’s total lifetime salary over a 30-year period?

SOLUTION The salary for the fi rst year is $20,000. With a 5% raise, the second-year salary is computed as follows:

Salary for year 2 = 20,000 + 20,000(0.05) = 20,000(1 + 0.05) = 20,000(1.05).

Each year, the salary is 1.05 times what it was in the previous year. Thus, the salary for year 3 is 1.05 times 20,000(1.05), or 20,000(1.05)2. The salaries for the fi rst fi ve years are given in the table.

TECHNOLOGY To fi nd

a10

i=1 6 # 2i

on a graphing utility, enter

� SUM � � SEQ �(6 * 2x, x, 1, 10, 1).

Then press � ENTER � .

Yearly Salaries

Year 1 Year 2 Year 3 Year 4 Year 5 N

20,000 20,000(1.05) 20,000(1.05)2 20,000(1.05)3 20,000(1.05)4 c

The numbers in the bottom row form a geometric sequence with a1 = 20,000 and r = 1.05. To fi nd the total salary over 30 years, we use the formula for the sum of the fi rst n terms of a geometric sequence, with n = 30.

Sn =a1(1 - rn)

1 - r

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

1-1.05

Total salaryover 30 years

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

-0.05

� 1,328,777 Use a calculator.

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



Check Point 6 A job pays a salary of $30,000 the fi rst year. During the next 29 years, the salary increases by 6% each year. What is the total lifetime salary over the 30-year period?

Annuities The compound interest formula

A = P(1 + r)t

gives the future value, A, after t years, when a fi xed amount of money, P, the principal, is deposited in an account that pays an annual interest rate r (in decimal form) compounded once a year. However, money is often invested in small amounts at 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 until you retire. An annuity is a sequence of equal payments made at equal time periods. An IRA is an example of an annuity.

Suppose P dollars is deposited into an account at the end of each year. The account pays an annual interest rate, r, compounded annually. At the end of the fi rst year, the account contains P dollars. At the end of the second year, P dollars is deposited again. At the time of this deposit, the fi rst deposit has received interest earned during the second year. The value of the annuity is the sum of all deposits made plus all interest paid. Thus, the value of the annuity after two years is

P+P(1+r).

First-year depositof P dollars withinterest earned for

a year

Deposit of Pdollars at end of

second year

The value of the annuity after three years is

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


third year

The value of the annuity after t years is

P+P(1+r)+P(1+r)2+P(1+r)3+Á+P(1+r)t–1.

First-year depositof P dollars withinterest earnedover t − 1 years


year t

This is the sum of the terms of a geometric sequence with fi rst term P and common ratio 1 + r. We use the formula

Sn =a1(1 - rn)

1 - r

to fi nd the sum of the terms:

St =P[1 - (1 + r)t]

1 - (1 + r)=

P[1 - (1 + r)t]-r

=P[(1 + r)t - 1]

r.

This formula gives the value of an annuity after t years if interest is compounded once a year. We can adjust the formula to fi nd the value of an annuity if equal payments are made at the end of each of n yearly compounding periods.

� Find the value of an annuity.



EXAMPLE 7 Determining the Value of an Annuity

At age 25, to save for retirement, you decide to deposit $200 at the end of each month 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 you

retire at 65 is its value after 40 years.

A =PJ a1 +

rnbnt

- 1Rrn

Use the formula for the value of an annuity.

A =200J a1 +

0.07512b12 #40

- 1R0.075

12

The annuity involves month-end deposits of $200: P = 200. The interest rate is 7.5%: r = 0.075. The interest is compounded monthly: n = 12. The number of years is 40: t = 40.

Value of an Annuity: Interest Compounded n Times per Year

If P is the deposit made at the end of each compounding period for an annuity at r percent annual interest compounded n times per year, the value, A, of the annuity after t years is

A =PJ a1 +

rnbnt

- 1Rrn

.

Blitzer Bonus Stashing Cash and

Making Taxes Less Taxing

As you prepare for your future career, retirement probably seems very far away. Making regular deposits into an IRA may not be fun, but there is a special incentive from Uncle Sam that makes it far more appealing. Traditional IRAs are tax-deferred savings plans . This means that you do not pay taxes on deposits and interest until you begin withdrawals, typically at retirement. Before then, yearly deposits count as adjustments to gross income and are not part of your taxable income. Not only do you get a tax break now, but also you ultimately earn more. This is because you do not pay taxes on interest from year to year, allowing earnings to accumulate until you start withdrawals. With a tax code that encourages long-term savings, opening an IRA early in your career is a smart way to gain more control over how you will spend a large part of your life.

=200[(1 + 0.00625)480 - 1]

0.00625

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

=200[(1.00625)480 - 1]

0.00625

�200(19.8989 - 1)

0.00625

Use a calculator to fi nd (1.00625)480:

1.00625 � yx � 480 � = � .

� 604,765

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

b. ≠$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 fi ve times the amount of your contributions to the IRA. ● ● ●



Check Point 7 At age 30, to save for retirement, you decide to deposit $100 at the end 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 Series An infi nite sum of the form

a1 + a1r + a1r2 + a1r

3 + g + a1rn-1 + g

with fi rst term a1 and common ratio r is called an infi nite geometric series . How can we determine which infi nite geometric series have sums and which do not? We look at what happens to rn as n gets larger in the formula for the sum of the fi rst n terms of this series, namely,

Sn =a1(1 - rn)

1 - r.

If r is any number between -1 and 1, that is, -1 6 r 6 1, the term rn approaches 0 as n gets larger. For example, consider what happens to rn for r = 1

2 :

12

12

=

These numbers are approaching 0 as n gets larger.

a b1 12

14

=a b2 12

18

=a b3 12

116

=a b4 12

132

=a b5 12

164

= .a b6

Take another look at the formula for the sum of the fi rst n terms of a geometric sequence.

Sn=a1(1-rn)

1-rIf −1 < r < 1,

rn approaches 0 as n gets larger.

Let us replace rn with 0 in the formula for Sn. This change gives us a formula for the sum of an infi nite geometric series with a common ratio between -1 and 1.

� Use the formula for the sum of an infi nite geometric series.

The Sum of an Infi nite Geometric Series

If -1 6 r 6 1 (equivalently, |r | 6 1 ), then the sum of the infi nite geometric series

a1 + a1r + a1r2 + a1r

3 + g ,

in which a1 is the fi rst term and r is the common ratio, is given by

S =a1

1 - r.

If � r � Ú 1, the infi nite series does not have a sum.



To use the formula for the sum of an infi nite geometric series, we need to know the fi rst term and the common ratio. For example, consider

12

14

+18

+116

+132

+ +Á .First term, a1, is .

Common ratio, r, is .

12

r = ÷ = � 2 =14

14

12

12

a2a1

With r =12

, the condition that |r | 6 1 is met, so the infi nite geometric series has a

sum given by S =a1

1 - r. The sum of the series is found as follows:

12

+14

+18

+116

+132

+ g =a1

1 - r=

12

1 -12

=

1212

= 1.

Thus, the sum of the infi nite geometric series is 1. Notice how this is illustrated in Figure 11.7 . As more terms are included, the sum is approaching the area of one complete circle.

EXAMPLE 8 Finding the Sum of an Infi nite Geometric Series

Find the sum of the infi nite geometric series: 38 - 316 + 3

32 - 364 + g.

SOLUTION Before fi nding the sum, we must fi nd the common ratio.

r =a2

a1=

-31638

= - 316

# 83= -

12

Because r = - 12, the condition that |r | 6 1 is met. Thus, the infi nite geometric series has a sum.

S =a1

1 - r This is the formula for the sum of an infi nite

geometric series. Let a1 =38

and r = - 12

.

=

38

1 - a- 12b

=

3832

=38# 23=

14

Thus, the sum of 38 - 316 + 3

32 - 364 + g is 14. Put in an informal way, as we continue

to add more and more terms, the sum is approximately 14. ● ● ●

Check Point 8 Find the sum of the infi nite geometric series: 3 + 2 + 4

3 + 89 + g.

qq

qq~

~

~

~

Ω

ΩΩ

116

116

132

FIGURE 11.7 The sum 12 + 1

4 + 18 + 1

16 + 132 + g is

approaching 1.



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

EXAMPLE 9 Writing a Repeating Decimal as a Fraction

Express 0.7 as a fraction in lowest terms.

SOLUTION

0.7 = 0.7777 . . . =710

+7

100+

71000

+7

10,000+ g

Observe that 0.7 is an infi nite geometric series with fi rst term 710 and common ratio 110. Because r = 1

10, the condition that � r � 6 1 is met. Thus, we can use our formula to fi nd the sum. Therefore,

0.7 =a1

1 - r=

710

1 -110

=

710910

=710

# 109

=79

.

An equivalent fraction for 0.7 is 79. ● ● ●

Check Point 9 Express 0.9 as a fraction in lowest terms.

Infi nite geometric series have many applications, as illustrated in Example 10.

EXAMPLE 10 Tax Rebates and the Multiplier Effect

A tax rebate that returns a certain amount of money to taxpayers can have a total effect on the economy that is many times this amount. In economics, this phenomenon is called the multiplier effect . Suppose, for example, that the government reduces taxes so that each consumer has $2000 more income. The government assumes that each person will spend 70% of this (= +1400). The individuals and businesses receiving this $1400 in turn spend 70% of it (= +980), creating extra income for other people to spend, and so on. Determine the total amount spent on consumer goods from the initial $2000 tax rebate.

SOLUTION

The total amount spent is given by the infi nite geometric series

1400+980+686+Á .

70% of1400

70% of980

The fi rst term is 1400: a1 = 1400. The common ratio is 70%, or 0.7: r = 0.7. Because r = 0.7, the condition that |r | 6 1 is met. Thus, we can use our formula to fi nd the sum. Therefore,

1400 + 980 + 686 + g = a1

1 - r=

14001 - 0.7

� 4667.

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

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

$1400

$980

$686

70% is spent.

70% is spent.



1. A sequence in which each term after the fi rst is obtained by multiplying the preceding term by a fi xed nonzero constant is called a/an sequence. The amount by which we multiply each time is called the of the sequence.

2. The n th term of the sequence described in Exercise 1is given by the formula an = , where a1 is the and r is the of the sequence.

3. The sum, Sn, of the fi rst n terms of the sequence described in Exercise 1 is given by the formula Sn = , where a1 is the and r is the , r � 1.

4. A sequence of equal payments made at equal time periods is called a/an . Its value, A, after t years is given by the formula

A =P[(1 + r

n)nt - 1]rn

,

where is the deposit made at the end of each compounding period at percent annual interest compounded times per year.

5. An infi nite sum of the form a1 + a1r + a1r

2 + a1r3 + g

is called a/an .If -1 6 r 6 , its sum, S, is given by the

formula S = . The series does not have a sum if .

6. The fi rst four terms of a6

i=12i are , ,

, and . The common ratio is.

Determine whether each sequence is arithmetic or geometric . 7. 4, 8, 12, 16, 20, . . . 8. 4, 8, 16, 32, 64, . . . 9. 1, -3, 9, -27, 81, . . .

10. -1, 1, 3, 5, 7, . . .

Fill in each blank so that the resulting statement is true .

CONCEPT AND VOCABULARY CHECK

Practice Exercises In Exercises 1–8, write the fi rst fi ve terms of each geometric sequence.

1. a1 = 5, r = 3 2. a1 = 4, r = 3 3. a1 = 20, r = 1

2 4. a1 = 24, r = 13

5. an = -4an-1, a1 = 10 6. an = -3an-1, a1 = 10 7. an = -5an-1, a1 = -6 8. an = -6an-1, a1 = -2

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to fi nd the indicated term of each sequence with the given fi rst term, a1, and common ratio, r.

9. Find a8 when a1 = 6, r = 2. 10. Find a8 when a1 = 5, r = 3. 11. Find a12 when a1 = 5, r = -2. 12. Find a12 when a1 = 4, r = -2. 13. Find a40 when a1 = 1000, r = - 12. 14. Find a30 when a1 = 8000, r = - 12. 15. Find a8 when a1 = 1,000,000, r = 0.1. 16. Find a8 when a1 = 40,000, r = 0.1.

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for an to fi nd a7, the seventh term of the sequence.

17. 3, 12, 48, 192, c 18. 3, 15, 75, 375, c 19. 18, 6, 2, 23, . . . . 20. 12, 6, 3, 32, . . . .

21. 1.5, -3, 6, -12, . . . 22. 5, -1, 15, - 125, . . . .

23. 0.0004, -0.004, 0.04, -0.4, . . . 24. 0.0007, -0.007, 0.07, -0.7,c

Use the formula for the sum of the fi rst n terms of a geometric sequence to solve Exercises 25–30.

25. Find the sum of the fi rst 12 terms of the geometric sequence: 2, 6, 18, 54, . . . .

26. Find the sum of the fi rst 12 terms of the geometric sequence: 3, 6, 12, 24, . . . .

27. Find the sum of the fi rst 11 terms of the geometric sequence: 3, -6, 12, -24, . . . .

28. Find the sum of the fi rst 11 terms of the geometric sequence: 4, -12, 36, -108, . . . .

29. Find the sum of the fi rst 14 terms of the geometric sequence: - 32, 3, -6, 12, . . . .

30. Find the sum of the fi rst 14 terms of the geometric sequence: - 1

24, 112, - 16, 13, . . . .

In Exercises 31–36, fi nd the indicated sum. Use the formula for the sum of the fi rst n terms of a geometric sequence.

31. a8

i=1 3i 32. a

6

i=1 4i 33. a

10

i=1 5 # 2i

34. a7

i=1 4(-3)i 35. a

6

i=1 11

22i+1 36. a6

i=1 11

32i+1

EXERCISE SET 11.3



Application Exercises Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises 65–68.

In Exercises 65–66, suppose you save $1 the fi rst day of a month, $2 the second day, $4 the third day, and so on. That is, each day you save twice as much as you did the day before.

65. What will you put aside for savings on the fi fteenth day of the month?

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

67. A professional baseball player signs a contract with a beginning salary of $3,000,000 for the fi rst year and an annual increase of 4% per year beginning in the second year. That is, beginning in year 2, the athlete’s salary will be 1.04 times what it was in the previous year. What is the athlete’s salary for year 7 of the contract? Round to the nearest dollar.

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

In Exercises 69–70, you will develop geometric sequences that model the population growth for California and Texas, the two most-populated U.S. states.

69. The table shows the population of California for 2000 and 2010, with estimates given by the U.S. Census Bureau for 2001 through 2009.

In Exercises 37–44, fi nd the sum of each infi nite geometric series.

37. 1 +13

+19

+1

27+ g 38. 1 +

14

+1

16+

164

+ g

39. 3 +34

+3

42 +3

43 + g 40. 5 +56

+5

62 +5

63 + g

41. 1 -12

+14

-18

+ g 42. 3 - 1 +13

-19

+ g

43. a�

i=1 8(-0.3)i-1 44. a

�

i=1 12(-0.7)i-1

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms.

45. 0.5 =5

10+

5100

+5

1000+

510,000

+ g

46. 0.1 =1

10+

1100

+1

1000+

110,000

+ g

47. 0.47 =47

100+

4710,000

+47

1,000,000+ g

48. 0.83 =83

100+

8310,000

+83

1,000,000+ g

49. 0.257 50. 0.529

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, fi nd the common difference; if it is geometric, fi nd the common ratio.

51. an = n + 5 52. an = n - 3

53. an = 2n 54. an = 1122n

55. an = n2 + 5 56. an = n2 - 3

Practice Plus In Exercises 57–62, let

{an} = -5, 10, -20, 40, . . . ,

{bn} = 10, -5, -20, -35, . . . ,

and

{cn} = -2, 1, - 12, 14, . . . .

57. Find a10 + b10. 58. Find a11 + b11. 59. Find the difference between the sum of the fi rst 10 terms of

{an} and the sum of the fi rst 10 terms of {bn}. 60. Find the difference between the sum of the fi rst 11 terms of

{an} and the sum of the fi rst 11 terms of {bn}. 61. Find the product of the sum of the fi rst 6 terms of {an} and the

sum of the infi nite series containing all the terms of {cn}. 62. Find the product of the sum of the fi rst 9 terms of {an} and the

sum of the infi nite series containing all the terms of {cn}.

In Exercises 63–64, fi nd a2 and a3 for each geometric sequence.

63. 8, a2, a3, 27 64. 2, a2, a3, -54

Year 2000 2001 2002 2003 2004 2005

Population in millions 33.87 34.21 34.55 34.90 35.25 35.60

Year 2006 2007 2008 2009 2010

Population in millions 36.00 36.36 36.72 37.09 37.25

a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric.

b. Write the general term of the geometric sequence modeling California’s population, in millions, n years after 1999.

c. Use your model from part (b) to project California’s population, in millions, for the year 2020. Round to two decimal places.

70. The table shows the population of Texas for 2000 and 2010, with estimates given by the U.S. Census Bureau for 2001 through 2009.

Year 2000 2001 2002 2003 2004 2005

Population in millions 20.85 21.27 21.70 22.13 22.57 23.02

Year 2006 2007 2008 2009 2010

Population in millions 23.48 23.95 24.43 24.92 25.15



78. To save money for a sabbatical to earn a master’s degree, you deposit $2500 at the end of each year in an annuity that pays 6.25% compounded annually.

a. How much will you have saved at the end of fi ve years? b. Find the interest.

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



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



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

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


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

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


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

a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric.

b. Write the general term of the geometric sequence modeling Texas’s population, in millions, n years after 1999.

c. Use your model from part (b) to project Texas’s population, in millions, for the year 2020. Round to two decimal places.

Use the formula for the sum of the fi rst n terms of a geometric sequence to solve Exercises 71–76.

In Exercises 71–72, you save $1 the fi rst day of a month, $2 the second day, $4 the third day, continuing to double your savings each day.

71. What will your total savings be for the fi rst 15 days? 72. What will your total savings be for the fi rst 30 days? 73. A job pays a salary of $24,000 the fi rst year. During the next

19 years, the salary increases by 5% each year. What is the total lifetime salary over the 20-year period? Round to the nearest dollar.

74. You are investigating two employment opportunities. Company A offers $30,000 the fi rst year. During the next four years, the salary is guaranteed to increase by 6% per year. Company B offers $32,000 the fi rst year, with guaranteed annual increases of 3% per year after that. Which company offers the better total salary for a fi ve-year contract? By how much? Round to the nearest dollar.

75. A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is 90% of the previous length.

20, 0.9(20), 0.92(20), 0.93(20), Á

1stswing

2ndswing

3rdswing

4thswing

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

76. A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is 96% of the previous length.

16, 0.96(16), (0.96)2(16), (0.96)3(16), Á

1stswing

2ndswing

3rdswing

4thswing

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

Use the formula for the value of an annuity to solve Exercises 77–84. Round answers to the nearest dollar.

77. To save money for a sabbatical to earn a master’s degree, you deposit $2000 at the end of each year in an annuity that pays 7.5% compounded annually.

a. How much will you have saved at the end of fi ve years? b. Find the interest.

Lump-Sum Deposit Rate Time

$30,000 5% compounded annually

20 years

Periodic Deposits Rate Time

$1500 at the end of each year

5% compounded annually

20 years

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

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

Lump-Sum Deposit Rate Time

$40,000 6.5% compounded annually

25 years

Periodic Deposits Rate Time

$1600 at the end of each year

6.5% compounded annually

25 years

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



In Exercises 99–100, use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of f and discuss its relationship to the sum of the given series.

99. Function Series

f(x) =2J1 - a1

3bxR

1 -13

2 + 2a13b + 2a1

3b2

+ 2a13b3

+ g

100. Function Series

f(x) =4[1 - (0.6)x]

1 - 0.6 4 + 4(0.6) + 4(0.6)2 + 4(0.6)3 + g

Critical Thinking Exercises Make Sense? In Exercises 101–104, determine whether each statement makes sense or does not make sense, and explain your reasoning.

101. There’s no end to the number of geometric sequences that I can generate whose fi rst term is 5 if I pick nonzero numbers r and multiply 5 by each value of r repeatedly.

102. I’ve noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

103. I modeled California’s population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

104. I used a formula to fi nd the sum of the infi nite geometric series 3 + 1 + 1

3 + 19 + g and then checked my answer by

actually adding all the terms.

In Exercises 105–108, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

105. The sequence 2, 6, 24, 120, c is an example of a geometric sequence.

106. The sum of the geometric series 12 + 14 + 1

8 + g+ 1512 can

only be estimated without knowing precisely what terms occur between 18 and 1

512.

107. 10 - 5 +52

-54

+ g =10

1 -12

108. If the nth term of a geometric sequence is an = 3(0.5)n-1, the common ratio is 12.

109. In a pest-eradication program, sterilized male fl ies are released into the general population each day. Ninety percent of those fl ies will survive a given day. How many fl ies should be released each day if the long-range goal of the program is to keep 20,000 sterilized fl ies in the population?

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

Use the formula for the sum of an infi nite geometric series to solve Exercises 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 by factory personnel. The people in the town who receive this money are expected to spend 60% of what they receive in the town, and so on. What is the total of all this spending, called the total economic 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 fi gure is continued indefi nitely, what fractional part of the largest square will eventually be shaded?

Writing in Mathematics 88. 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 fi nd the general term of a geometric sequence. 91. Explain how to fi nd the sum of the fi rst n terms of a geometric

sequence without having to add up all the terms. 92. What is an annuity? 93. What is the difference between a geometric sequence and an

infi nite geometric series? 94. How do you determine if an infi nite geometric series has

a sum? Explain how to fi nd the sum of such an infi nite 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 day 5, and so on, for 30 days? Explain.

96. For the fi rst 30 days of a fl u outbreak, the number of students on your campus who become ill is increasing. Which is worse: The number of students with the fl u is increasing arithmetically or is increasing geometrically? Explain your answer.

Technology Exercises 97. Use the � SEQ � (sequence) capability of a graphing utility and

the formula you obtained for an to verify the value you found for a7 in any three exercises from Exercises 17–24.

98. Use the capability of a graphing utility to calculate the sum of a sequence to verify any three of your answers to Exercises 31–36.


Mid-Chapter Check Point 1055

Preview Exercises Exercises 112–114 will help you prepare for the material covered in the next section.

In Exercises 112–113, show that

1 + 2 + 3 + g+ n =n(n + 1)

2

is true for the given value of n.

112. n = 3: Show that 1 + 2 + 3 =3(3 + 1)

2.

113. n = 5: Show that 1 + 2 + 3 + 4 + 5 =5(5 + 1)

2.

114. Simplify: k(k + 1)(2k + 1)

6+ (k + 1)2.

Group Exercise 111. Group members serve as a fi nancial team analyzing the

three options given to the professional baseball player described in the chapter opener on page 1017. As a group, determine which option provides the most amount of money over the six-year contract and which provides the least. Describe one advantage and one disadvantage to each option.

WHAT YOU KNOW: We learned that a sequence is a function whose domain is the set of positive integers. In an arithmetic sequence, each term after the fi rst differs from the preceding term by a constant, the common difference, d. In a geometric sequence, each term after the fi rst is obtained by multiplying the preceding term by a nonzero constant, the common ratio, r. We found the general term of arithmetic sequences [an = a1 + (n - 1)d] and geometric sequences [an = a1r

n-1] and used these formulas to fi nd particular terms. We determined the sum of the fi rst n terms

of arithmetic sequences cSn =n2

(a1 + an) d and geometric

sequences JSn =a1(1 - rn)

1 - rR . Finally, we determined the

sum of an infi nite geometric series,

a1 + a1r + a1r2 + a1r

3 + g, if -1 6 r 6 1¢S =a1

1 - r≤.

In Exercises 1–4, write the fi rst fi ve terms of each sequence. Assume that d represents the common difference of an arithmetic sequence and r represents the common ratio of a geometric sequence.

1. an = (-1)n+1 n

(n - 1)! 2. a1 = 5, d = -3

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

In Exercises 5–7, write a formula for the general term (the nth term) of each sequence. Then use the formula to fi nd the indicated term.

5. 2, 6, 10, 14, . . . ; a20 6. 3, 6, 12, 24, . . . ; a10

7. 32

, 1, 12

, 0, . . . ; a30

8. Find the sum of the fi rst ten terms of the sequence:

5, 10, 20, 40, . . . .

9. Find the sum of the fi rst 50 terms of the sequence:

-2, 0, 2, 4, . . . .

10. Find the sum of the fi rst ten terms of the sequence:

-20, 40, -80, 160, . . . .

11. Find the sum of the fi rst 100 terms of the sequence:

4, -2, -8, -14, . . . .

In Exercises 12–15, fi nd each indicated sum.

12. a4

i=1 (i + 4)(i - 1) 13. a

50

i=1 (3i - 2)

14. a6

i=1 a3

2b i

15. a�

i=1 a-

25b i-1

16. Express 0.45 as a fraction in lowest terms. 17. Express the sum using summation notation. Use i for the

index of summation.

13

+24

+35

+ g + 1820

18. A skydiver falls 16 feet during the fi rst second of a dive, 48 feet during the second second, 80 feet during the third second, 112 feet during the fourth second, and so on. Find the distance that the skydiver falls during the 15th second and the total distance the skydiver falls in 15 seconds.

19. If the average value of a house increases 10% per year, how much will a house costing $120,000 be worth in 10 years? Round to the nearest dollar.

Mid-Chapter Check Point CHAPTER 11


section 11.3 geometric sequences and series · 9, -3, 1 , - 1 3, 1 9. r =-3 9 = - 1 3 figure 11.6...

Documents