Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

12.4: Geometric Sequences and Series

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives

By the end of this section, you will be able to:

  • Determine if a sequence is geometric
  • Find the general term (nth term) of a geometric sequence
  • Find the sum of the first n terms of a geometric sequence
  • Find the sum of an infinite geometric series
  • Apply geometric sequences and series in the real world

Before you get started, take this readiness quiz.

  1. Simplify: 2432.
    If you missed this problem, review Example 1.24.
  2. Evaluate: a. 34 b. (12)4.
    If you missed this problem, review Example 1.19.
  3. If f(x)=43x, find a. f(1) b. f(2) c. f(3).
    If you missed this problem, review Example 3.49.

Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.

Definition 12.4.1

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, anan1, is r, the common ratio. n is greater than or equal to two.

Consider these sequences.

This figure shows two sets of sequences where r is the common ratio.
Figure 12.3.1
Example 12.4.1

Determine if each sequence is geometric. If so, indicate the common ratio.

  1. 4,8,16,32,64,128,
  2. 2,6,12,36,72,216,
  3. 27,9,3,1,13,19,

Solution:

To determine if the sequence is geometric, we find the ratio of the consecutive terms shown.

a. Find the ratio of the consecutive terms

4,8,16,32,64,128,84168321664321286422222

The sequence is geometric. The common ration is r=2.

b. Find the ratio of the consecutive terms

2,6,12,36,72216,62126361272362167232323

The sequence is not geometric. There is no common ratio.

c. Find the ratio of the consecutive terms

27,9,3,1,13,19,927391313119131313131313

The sequence is geometric. The common ratio is r=13.

Exercise 12.4.1

Determine if each sequence is geometric. If so indicate the common ratio.

  1. 7,21,63,189,567,1,701,
  2. 64,16,4,1,14,116,
  3. 2,4,12,48,240,1,440,
Answer
  1. The sequence is geometric with common ratio r=3.
  2. The sequence is geometric with common ratio d=14.
  3. The sequence is not geometric. There is no common ratio.
Exercise 12.4.2

Determine if each sequence is geometric. If so indicate the common ratio.

  1. 150,30,15,5,52,0,
  2. 5,10,20,40,80,160,
  3. 8,4,2,1,12,14,
Answer
  1. The sequence is not geometric. There is no common ratio.
  2. The sequence is geometric with common ratio r=2.
  3. The sequence is geometric with common ratio r=12.

If we know the first term, a1, and the common ratio, r, we can list a finite number of terms of the sequence.

Example 12.4.2

Write the first five terms of the sequence where the first term is 3 and the common ratio is r=2.

Solution:

We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.

a1a2a3a4a533(2)6(2)12(2)24(2)6122448

Answer:

The sequence is 3,6,12,24,48,

Exercise 12.4.3

Write the first five terms of the sequence where the first term is 7 and the common ratio is r=3.

Answer

7,21,63,189,567

Exercise 12.4.4

Write the first five terms of the sequence where the first term is 6 and the common ratio is r=4.

Answer

6,24,96,384,1536

Find the General Term (nth Term) of a Geometric Sequence

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is a1 and the common ratio is r. We will then look for a pattern.

This figure shows an image of a geometric sequence.
Figure 12.3.2

As we look for a pattern in the five terms above, we see that each of the terms starts with a1.

The first term, a1, is not multiplied by any r. In the second term, the a1 is multiplied by r. In the third term, the a1 is multiplied by r two times (rr or r2). In the fourth term, the a1 is multiplied by r three times (rrr or r3) and in the fifth term, the a1 is multiplied by r four times. In each term, the number of times a1 is multiplied by r is one less than the number of the term. This leads us to the following

an=a1rn1

Definition 12.4.2

The general term of a geometric sequence with first term a1 and the common ratio r is

an=a1rn1

We will use this formula in the next example to find the fourteenth term of a sequence.

Example 12.4.3

Find the fourteenth term of a sequence where the first term is 64 and the common ratio is r=12.

Solution:

an=a1rn1

To find the fourteenth term, a14,use the formula with a1=64 and r=12.

a14=64(12)141

Substitute in the values.

a14=64(12)13

Simplify.

a14=1128

Exercise 12.4.5

Find the thirteenth term of a sequence where the first term is 81 and the common ratio is r=13.

Answer

16,561

Exercise 12.4.6

Find the twelfth term of a sequence where the first term is 256 and the common ratio is r=14.

Answer

116,384

Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.

Example 12.4.4

Find the twelfth term of the sequence 3,6,12,24,48,96, Find the general term for the sequence.

Solution:

To find the twelfth term, we use the formula, an=a1rn1, and so we need to first determine a1 and the common ratio r.

The first term is three.

3,6,12,24,48,96,
a1=3

Find the common ratio.

6312624124824964822222 The common ratio is r=2

To find the twelfth term, a12,use the formula with a1=3 and r=2.

an=a1rn1

Substitute in the values.

a12=32121

Simplify.

a12=3211
a12=6,144

Find the general term. We use the formula with a1=3 and r=2.

an=a1rn1
an=3(2)n1

Exercise 12.4.7

Find the ninth term of the sequence 6,18,54,162,486,1,458, Then find the general term for the sequence.

Answer

a9=39,366. The general term is an=6(3)n1.

Exercise 12.4.8

Find the eleventh term of the sequence 7,14,28,56,112,224, Then find the general term for the sequence.

Answer

a11=7,168. The general term is an=7(2)n1.

Find the Sum of the First n Terms of a Geometric Sequence

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, Sn, of the first n terms of a geometric sequence is written as Sn=a1+a2+a3++an. We can write this sum by starting with the first term, a1, and keep multiplying by r to get the next term as:

Sn=a1+a1r+a1r2++a1rn1

Let’s also multiply both sides of the equation by r.

rSn=a1r+a1r2+a1r3++a1rn

Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.

Sn=a1+a1r+a1r2+a1r3++a1rn1rSn=a1r+a1r2+a1r3++a1rn1+a1rnSnrSn=a1a1rn

We factor both sides.

Sn(1r)=a1(1rn)

To obtain the formula for Sn, divide both sides by (1r).

Sn=a1(1rn)1r

Definition 12.4.3

The sum, Sn, of the first n terms of a geometric sequence is

Sn=a1(1rn)1r

where a1 is the first term and r is the common ratio, and r is not equal to one.

We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.

Example 12.4.5

Find the sum of the first 20 terms of the geometric sequence 7,14,28,56,112,224,

Solution:

To find the sum, we will use the formula Sn=a1(1rn)1r. We know a1=7, r=2, and n=20,

Knowing a1=7,r=2, and n=20, use the sum formula.

Sn=a1(1rn)1r

Substitute in the values.

S20=7(1220)12

Simplify.

S20=7,340,025

Exercise 12.4.9

Find the sum of the first 20 terms of the geometric sequence 3,6,12,24,48,96,

Answer

3,145,725

Exercise 12.4.10

Find the sum of the first 20 terms of the geometric sequence 6,18,54,162,486,1,458,

Answer

10,460,353,200

In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.

To use the formula, we need r. We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form ki=1a(r)i, where r is the common ratio.

Example 12.4.6

Find the sum: 15i=12(3)i.

Solution:

To find the sum, we will use the formula Sn=a1(1rn)1r, which requires a1 and r. We will write out a few of the terms, so we can get the needed information.

  .
Write out the first few terms. .
Identify a1. .
Find the common ratio. .
Knowing a1=6, r=3, and n=15, use the sum formula. .
Substitute in the values. .
Simplify. .
Table 12.3.1
Exercise 12.4.11

Find the sum: 15i=16(2)i.

Answer

393,204

Exercise 12.4.12

Find the sum: 10i=15(2)i.

Answer

10,230

Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is a1 and common ratio is r and is written

a1+a1r+a1r2++a1rn1+

Definition 12.4.4

An infinite geometric series is an infinite sum whose first term is a1 and common ratio is r and is written

a1+a1r+a1r2++a1rn1+

We know how to find the sum of the first n terms of a geometric series using the formula, Sn=a1(1rn)1r. But how do we find the sum of an infinite sum?

Let’s look at the infinite geometric series 3+6+12+24+48+96+.. Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see a1=3 and r=2

Sn=a1(1rn)1rSn=a1(1rn)1rSn=a1(1rn)1rS10=3(1210)12S30=3(1230)12S50=3(1250)12S10=3,069S30=3,221,225,469S503.38×1015

As n gets larger and larger, the sum gets larger and larger. This is true when |r|1 and we call the series divergent. We cannot find a sum of an infinite geometric series when |r|1.

Let’s look at an infinite geometric series whose common ratio is a fraction less than one,
12+14+18+116+132+164+. Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see a1=12 and r=12.

Sn=a1(1rn)1rSn=a1(1rn)1rSn=a1(1rn)1rS10=12(11210)112S20=12(11220)112S30=12(11230)112S100.9990234375S200.9999990463S300.9999999991

Notice the sum gets larger and larger but also gets closer and closer to one. When |r|<1, the expression rn gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large), rn gets closer and closer to zero. In our sum formula, we can replace the rn with zero and then we get a formula for the sum, S, for an infinite geometric series when |r|<1.

Sn=a1(1rn)1rS=a1(10)1rS=a11r

This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in Sn as we are not adding a finite number of terms.

Definition 12.4.5

For an infinite geometric series whose first term is a1 and common ratio r,

If |r|<1,the sum is

S=a11r

If |r|1,the infinite geometric series does not have a sum. We say the series diverges.

Example 12.4.7

Find the sum of the infinite geometric series 54+18+6+2+23+29+

Solution:

To find the sum, we first have to verify that the common ratio |r|<1 and then we can use the sum formula S=a11r.

Find the common ratio.

r=1854r=618r=13r=13|r|<1

Identify a1.

a1=54

Knowing a1=54,r=13, use the sum formula.

S=a11r

Substitute in the values.

S=54113

Simplify.

S=81

Answer:

S=80

Exercise 12.4.13

Find the sum of the infinite geometric series 48+24+12+6+3+32+

Answer

96

Exercise 12.4.14

Find the sum of the infinite geometric series 64+16+4+1+14+116+

Answer

2563

An interesting use of infinite geometric series is to write a repeating decimal as a fraction.

Example 12.4.8

Write the repeating decimal 0.5 as a fraction.

Solution:

Rewrite the 0.5 showing the repeating five. Use place value to rewrite this as a sum. This is an infinite geometric series.

0.5555555555555
0.5+0.05+0.005+0.0005+

Find the common ratio.

r=0.050.5r=0.0050.05r=0.1r=0.1|r|<1

Identify a1

a1=0.5

Knowing a1=0.5,r=0.1, use the sum formula.

S=a11r

Substitute in the values.

S=0.510.1

Simplify.

S=0.50.9

Multiply the numerator and denominator by 10.

S=59

We are asked to find the fraction form.

0.5=59

Exercise 12.4.15

Write the repeating decimal 0.4 as a fraction.

Answer

49

Exercise 12.4.16

Write the repeating decimal 0.8 as a fraction.

Answer

89

Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.

Example 12.4.9

The government has decided to give a $1,000 tax rebate to each household in order to stimulate the economy. The government statistics say that each household will spend 80% of the rebate in goods and services. The businesses and individuals who benefited from that 80% will then spend 80% of what they received and so on. The result is called the multiplier effect. What is the total effect of the rebate on the economy?

Solution:

Every time money goes into the economy, 80% of it is spent and is then in the economy to be spent. Again, 80% of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

1000+1000(0.8)+1000(0.8)2+

Here the first term is 1,000,a1=1000. The common ratio is 0.8,r=0.8. We can evaluate this sum since 0.8<1. We use the formula for the sum on an infinite geometric series.

S=a11r

Substitute in the values, a1=1,000 and r=0.8.

S=1,00010.8

Evaluate.

S=5,000

Answer:

The total effect of the $1,000 received by each household will be a $5,000 growth in the economy.

Exercise 12.4.17

What is the total effect on the economy of a government tax rebate of $1,000 to each household in order to stimulate the economy if each household will spend 90% of the rebate in goods and services?

Answer

$10,000

Exercise 12.4.18

What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend 85% of the rebate in goods and services?

Answer

$3,333.33

We have looked at a compound interest formula where a principal, P, is invested at an interest rate, r, for t years. The new balance, A, is A=P(1+rn)nt when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.

An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let n=1. That means there is one deposit per year.

A=P(1+rn)nt Let n=1.A=P(1+r1)1t Simplify. A=P(1+r)t

Suppose P dollars is invested at the end of each year. One year later that deposit is worth P(1+r)1 dollars, and another year later it is worth P(1+r)2 dollars. After t years, it will be worth P(1+r)t dollars.

End of year 1 End of year 2 End of year 3
First Deposit P @ end of year 1 P Amount 1 year later P(1+r)1 Amount 2 years later P(1+r)2
2nd Deposit P @ end of year 2   P Amount 1 year later P(1+r)1
3rd Deposit P @ end of year 3     P
Table 12.3.2

After three years, the value of the annuity is

P plus P times the quantity 1 plus r in parentheses, to the first power, plus P times the quantity 1 plus r, in parentheses, squared. This equals the money deposited at the end of year three, plus the money deposited at the end of year two, plus the money deposited at the end of year 1.
Figure 12.3.10

This a sum of the terms of a geometric sequence where the first term is P and the common ratio is 1+r. We substitute these values into the sum formula. Be careful, we have two different uses of r. The r in the sum formula is the common ratio of the sequence. In this case, that is 1+r where r is the interest rate.

St=a1(1rt)1r Substitute in the values. St=P(1(1+r)t)1(1+r) Simplify. St=P(1(1+r)t)rSt=P((1+r)t1)r

Remember our premise was that one deposit was made at the end of each year.

We can adapt this formula for n deposits made per year and the interest is compounded n times a year.

Definition 12.4.6

For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

At=P((1+rn)nt1)rn

Example 12.4.10

New parents decide to invest $100 per month in an annuity for their baby daughter. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at her eighteenth birthday?

Solution:

To find the Annuity formula, At=P((1+rn)nt1)rn, we need to identify P,r,n, and t.

Identify P, the amount invested each month.

P=100

Identify r, the annual interest rate, in decimal form.

r=0.05

Identify n, the number of times the deposit will be made and the interest compounded each year.

n=12

Identify t, the number of years.

t=18

Knowing P=100,r=0.05,n=12 and t=18, use the sum formula.

At=P((1+rn)nt1)rn

Substitute in the values.

At=100((1+0.0512)12.181)0.0512

Use the calculator to evaluate. Be sure to use parentheses as needed.

At=34.920.20

Answer:

The child will have $34,920.20

Exercise 12.4.19

New grandparents decide to invest $200 per month in an annuity for their grandson. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Answer

$88,868.36

Exercise 12.4.20

Arturo just got his first full-time job after graduating from college at age 27. He decided to invest $200 per month in an IRA (an annuity). The interest on the annuity is 8%, which is compounded monthly. How much will be in the Arturo’s account when he retires at his sixty-seventh birthday?

Answer

$698,201.57

Access these online resources for additional instruction and practice with sequences.

Key Concepts

  • General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term a1 and the common ratio r is

    an=a1rn1

  • Sum of the First n Terms of a Geometric Series: The sum, Sn, of the n terms of a geometric sequence is

    Sn=a1(1rn)1r

    where a1 is the first term and r is the common ratio.Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is a1 and common ratio is r and is written

    a1+a1r+a1r2++a1rn1+

  • Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is a1 and common ratio r,
    If |r|<1,the sum is

S=a11r

We say the series converges.

If |r|1, the infinite geometric series does not have a sum. We say the series diverges.

  • Value of an Annuity with Interest Compounded n Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

    At=P((1+rn)nt1)rn

Glossary

annuity
An annuity is an investment that is a sequence of equal periodic deposits.
common ratio
The ratio between consecutive terms in a geometric sequence, anan1, is r, the common ratio, where r greater than or equal to two.
geometric sequence
A geometric sequence is a sequence where the ratio between consecutive terms is always the same
infinite geometric series
An infinite geometric series is an infinite sum infinite geometric sequence.

This page titled 12.4: Geometric Sequences and Series is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?