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Geometric Sequences

Author: Sophia

what's covered

In this lesson, you will explore geometric sequences, which are sequences in which the ratio of any two consecutive terms is the same. Specifically, this lesson will cover:

1. Identifying Geometric Sequences
2. Finding Terms of a Geometric Sequence
3. Writing the Formula for the nth Term of a Geometric Sequence

1. Identifying Geometric Sequences

try it

Consider the sequence 5, 10, 20, 40, 80, ….

Can you guess the next three terms of the sequence?

It appears that each term is twice the term before it. Therefore, the next three terms appear to be 160, 320, and 640.

In the previous example, notice that each term is twice its previous term.

This is an example of a geometric sequence. A geometric sequence is a sequence in which every term after the first term is obtained by multiplying some constant to the previous term. This constant is called the common ratio, which is denoted r.

The sequence 5, 10, 20, 40, 80, … from above has common ratio r equals 2. When taking any two consecutive terms and dividing the second by the first, the ratio is equal to the common ratio.

There are two ways to determine if a sequence is geometric:

For every pair of consecutive terms, divide the second by the first. If all the ratios are the same, then the sequence is geometric. If not, then the sequence is not geometric.
Determine if the same number is multiplied to get from one term to the next. If so, then the sequence is geometric. If not, then the sequence is not geometric. This is harder to see sometimes, so using the first criteria is often more helpful.

EXAMPLE

Consider the sequence 2, -6, 18, -54, ….

This is a geometric sequence since each term is multiplied by -3 to get the next term. Therefore, the common ratio is r equals short dash 3.

Another way to check this is to compute the ratios between two consecutive terms:

$fraction numerator short dash 6 over denominator 2 end fraction equals short dash 3$ $fraction numerator 18 over denominator short dash 6 end fraction equals short dash 3$ $fraction numerator short dash 54 over denominator 18 end fraction equals short dash 3$

Since all ratios are -3, this is also the common ratio, and the sequence is geometric.

EXAMPLE

Consider the sequence 2, 6, 12, 36, 108, ….

This sequence is not geometric since 6 over 2 equals 3

, but

There is no common ratio.

try it

Consider the following sequences:

$fraction numerator short dash 1 over denominator 4 end fraction comma$ $fraction numerator short dash 1 over denominator 8 end fraction comma$ ...
27, 18, 12, 8, ...
1, -1, 1, -1, …

Determine if each sequence is geometric. If it is geometric, state its common ratio.

The sequence a subscript n

is not geometric. The sequence b subscript n

is geometric with common ratio r equals 2 over 3.

The sequence c subscript n

is geometric with common ratio r equals short dash 1.

terms to know

Geometric Sequence

A sequence that is obtained by selecting the first term, then multiplying by a constant to get each subsequent term.

Common Ratio

The constant ratio between two consecutive terms in a geometric sequence.

2. Finding Terms of a Geometric Sequence

When given the first term and a common ratio, a geometric sequence is formed by multiplying by the common ratio r to get each subsequent term.

EXAMPLE

A geometric sequence is to have first term a subscript 1 equals 20

and common ratio r equals 1 half.

Write the next three terms of the sequence.

The common ratio r equals 1 half

tells us that we multiply each term by 1 half

to get the next term.

Then, the next three terms are:

The geometric sequence is 20, 10, 5, 5 over 2 comma

….

A similar idea can be used to relate two terms of an geometric sequence with common ratio r.

EXAMPLE

A geometric sequence has term a subscript 2 equals 64

and common ratio r equals 3 over 4.

Find the value of a subscript 6.

For this example, consider this picture:

From the picture, we can see the following:

	Starting at multiply the common ratio 4 times to get the value of
	Substitute and
	Simplify.

Thus,

We can also use this strategy to find a missing common ratio if two terms are known.

EXAMPLE

A geometric sequence has terms a subscript 4 equals 108

and

What is the common ratio r, and what is the value of a subscript 9 ?

Using the given terms of the sequence:

	and are 3 spaces apart in the sequence; therefore, multiply by r three times to get the value of
	Substitute and
	Divide both sides by 108.
	Take the cube root of both sides.

Thus, the common ratio for the sequence is r equals short dash 1 third.

Next, find the value of a subscript 9.

	Since is known, relate it to which is 2 spaces after in the sequence.
	Substitute and
	Simplify.

Thus,

a subscript 9 equals short dash 4 over 9.

Note: since a subscript 4

is also known, we could have used the equation a subscript 9 equals a subscript 4 times r to the power of 5 comma

which gives the same results.

try it

Consider a geometric sequence with common ratio r and you are given the values of a subscript 2

and

Write a₇ in terms of a₂ and r.

a subscript 7 equals a subscript 2 r to the power of 5

Given a₂ = 5 and a₇ = 12.4416, find the value of r.

Replace

and

into the equation a subscript 7 equals a subscript 2 r to the power of 5 comma

then solve for r.

	Replace and into the equation.
	Divide both sides by 5.
	Take the 5th root of both sides.
	Simplify.

This means that the common ratio is r equals 1.2.

3. Writing the Formula for the nth Term of a Geometric Sequence

The recursive formula for a geometric sequence with first term a subscript 1 is a subscript n equals r times a subscript n minus 1 end subscript for n greater or equal than 2.

It is also possible to find an explicit formula for the nth term, which is always more desirable.

Suppose a geometric sequence has first term a subscript 1 and we wish to find a formula for a subscript n.

The picture below shows the first six terms of a geometric sequence, and how r is multiplied to get each subsequent term.

From the sequence, we have the following relationships between a subscript 1 comma r, and the other terms of the sequence:

Equation	Relationship
	Multiply r to to get the value of
	Multiply r twice to to get the value of
	Multiply r three times to to get the value of
	Multiply r four times to to get the value of
	Multiply r five times to to get the value of

This pattern continues.

Notice that the number of times a subscript 1 is multiplied by r is one less than the position of the term we seek.

This leads to the following formula, which is the most conventional way to find the nth term of a geometric sequence.

formula to know

nth Term of an Geometric Sequence

a subscript n equals a subscript 1 times r to the power of n minus 1 end exponent

EXAMPLE

Find a formula for the nth term: 400, 440, 484, …

Notice that the sequence is not arithmetic since the terms do not increase by the same amount.

Next, try computing ratios of consecutive terms.

Since 440 over 400 equals 1.1

and

we assume that this sequence is geometric with common ratio r equals 1.1.

Then, the nth term of the sequence is a subscript n equals 400 open parentheses 1.1 close parentheses to the power of n minus 1 end exponent.

EXAMPLE

Find a formula for the nth term: 27, -18, 12, -8, …

Notice that the sequence is not arithmetic since the terms do not increase by the same amount.

Next, try computing ratios of consecutive terms.

$fraction numerator short dash 18 over denominator 27 end fraction equals short dash 2 over 3$ $fraction numerator 12 over denominator short dash 18 end fraction equals short dash 2 over 3$ $fraction numerator short dash 8 over denominator 12 end fraction equals short dash 2 over 3$

Since all ratios are the same, assume this is a geometric sequence with common ratio r equals short dash 2 over 3.

Then, the formula for the nth term is a subscript n equals 27 open parentheses short dash 2 over 3 close parentheses to the power of n minus 1 end exponent.

a subscript n equals 27 open parentheses short dash 2 over 3 close parentheses to the power of n minus 1 end exponent.

watch

This video will walk you through the steps to find a formula for the nth term of -24, -36, -54, ….

try it

Consider the sequence 135, 45, 15, 5, ….

Does the sequence have a common difference? If so, what is it?

If the sequence is arithmetic, there should be a common difference between any two consecutive terms.

a subscript 2 minus a subscript 1 equals 45 minus 135 equals short dash 90

a subscript 3 minus a subscript 2 equals 15 minus 45 equals short dash 30

Since these differences are not equal, there is no common difference. Therefore, the sequence is not arithmetic.

Does the sequence have a common ratio? If so, what is it?

If the sequence is geometric, there should be a common ratio between any two consecutive terms.

a subscript 2 over a subscript 1 equals 45 over 135 equals 1 third

a subscript 3 over a subscript 2 equals 15 over 45 equals 1 third

a subscript 4 over a subscript 3 equals 5 over 15 equals 1 third

Since these ratios are equal, there is a common ratio, and its value is r equals 1 third.

Based on the first 4 terms, the sequence is geometric.

Write a formula for the nth term of the sequence.

With

and

the formula for the nth term of the sequence is a subscript n equals 135 open parentheses 1 third close parentheses to the power of n minus 1 end exponent.

a subscript n equals 135 open parentheses 1 third close parentheses to the power of n minus 1 end exponent.

Geometric sequences can be used in applied situations that involve a percent increase or decrease.

EXAMPLE

The starting salary at a new job is $60,000 and is expected to increase by 2% each year.

Write a formula for the salary during the nth year.
Estimate the salary during the 9th year.

To find the formula, recall that a 2% increase means that the previous salary will be multiplied by 102%, or 1.02, to obtain the salary in the next year.

This means that the salary follows a geometric sequence with a subscript 1 equals 60000

and

Then, the formula for the salary during the nth year is a subscript n equals 60000 open parentheses 1.02 close parentheses to the power of n minus 1 end exponent.

a subscript n equals 60000 open parentheses 1.02 close parentheses to the power of n minus 1 end exponent.

Then, the salary during the 9th year is a subscript 9 equals 60000 open parentheses 1.02 close parentheses to the power of 9 minus 1 end exponent almost equal to $ 70 comma 300

a subscript 9 equals 60000 open parentheses 1.02 close parentheses to the power of 9 minus 1 end exponent almost equal to $ 70 comma 300

to the nearest whole dollar.

try it

A high-tech photocopier for an office building is valued at $120,000 and retains 80% of its value from year to year.

Write a formula for the value of the photocopier in its nth year of use.

V subscript n equals 120000 open parentheses 0.8 close parentheses to the power of n minus 1 end exponent

To the nearest whole dollar, what is the value of the photocopier after 7 years?

V subscript 7 equals 120000 open parentheses 0.8 close parentheses to the power of 7 minus 1 end exponent comma

which is approximately $31,457.

summary

In this lesson, you learned how to identify geometric sequences, which are sequences in which every term after the first term is obtained by multiplying some constant, called the common ratio (r), to the previous term. You also learned how to find terms of a geometric sequence when given the first term and a common ratio, by multiplying by the common ratio r to get each subsequent term. Finally, you learned how to write the formula for the nth term of a geometric sequence, whether expressed recursively or explicitly (with preference being the explicit form).

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

Terms to Know

Common Ratio: The constant ratio between two consecutive terms in a geometric sequence.
Geometric Sequence: A sequence that is obtained by selecting the first term, then multiplying by a constant to get each subsequent term.

Formulas to Know

nth Term of an Geometric Sequence: $a subscript n equals a subscript 1 times r to the power of n minus 1 end exponent$

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Geometric Sequences

Table of Contents

1. Identifying Geometric Sequences

2. Finding Terms of a Geometric Sequence

3. Writing the Formula for the nth Term of a Geometric Sequence