Geometric Sequence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Sequence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio $r$ .

Multiply by the same number each step — 2, 6, 18, 54,... (multiply by 3). This is exponential growth.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A geometric sequence multiplies by a fixed common ratio $r$ every term, giving exponential growth or decay.

Common stuck point: The procedure for geometric sequence is the easy part; the trap is using $a_1 r^{n}$ instead of $a_1 r^{n-1}$ . Asking "Do I get the same number every time I divide a term by the one before it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I get the same number every time I divide a term by the one before it?

Worked Examples

Example 1

easy

A geometric sequence has

a_1 = 5

and

r = 2

. Find

a_8

and

S_8

Answer

a_8 = 640

;

S_8 = 1275

First step

Use the geometric sequence formula

a_n = a_1 \cdot r^{n-1}

with

a_1 = 5

r = 2

n = 8

Full solution

2
Find the 8th term: $a_8 = 5 \cdot 2^7 = 5 \cdot 128 = 640$
3
Apply the partial sum formula: $S_8 = a_1 \cdot \frac{1-r^8}{1-r} = 5 \cdot \frac{1-256}{1-2} = 5 \cdot 255 = 1275$

Geometric sequences grow exponentially. The formula

a_n = a_1 r^{n-1}

uses exponent

n-1

because the first term has no multiplications applied.

Example 2

medium

Bacteria double every 3 hours. Starting with 500, how many after 15 hours?

Example 3

medium

Find the 6th term of the geometric sequence: 2, 6, 18, ...

Example 4

medium

Find the sum

S_6

of the geometric series

4 + 12 + 36 + \ldots

Example 5

medium

A ball is dropped from

80

cm and bounces to

70\%

of its previous height each time. How high after the 4th bounce?

Example 6

hard

Evaluate the infinite series

\sum_{n=1}^{\infty} 6 \cdot \left(\tfrac{1}{3}\right)^{n-1}

Example 7

hard

Express the repeating decimal

0.\overline{36}

as a fraction using an infinite geometric series.

Example 8

challenge

A square of side

1

has another square inscribed by joining the midpoints of its sides, then another inside that, and so on. Find the total area of all squares in the infinite nesting.

Example 9

challenge

A drug clears the body so that

20\%

remains after each 6-hour interval. A patient takes 100 mg every 6 hours starting at

t = 0

. What is the long-run steady-state amount in the body just after a dose?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

r

and

a_6

for the sequence

3, 6, 12, 24, \ldots

Example 2

hard

In a geometric sequence,

a_2 = 6

and

a_5 = 48

. Find

a_1

and

r

Example 3

easy

Find the common ratio of

3, 6, 12, 24, \ldots

Example 4

easy

Find the 5th term of a geometric sequence with

a_1=2

and

r=3

Example 5

easy

2, 6, 18, 54

arithmetic or geometric?

Example 6

easy

The first term is 5 and

r=2

. List the first four terms.

Example 7

easy

With

a_1=4

and

r=-2

, find the first four terms.

Example 8

easy

Find the sum of the first 4 terms of

3, 6, 12, 24

Example 9

easy

Write the general term for

2, 6, 18, 54, \ldots

Example 10

easy

48, 24, 12, 6, \ldots

, find the common ratio.

Example 11

medium

The 2nd term of a geometric sequence is 12 and the 4th term is 48. Find

r

(positive) and

a_1

Example 12

medium

Find the sum of the first 6 terms of

1, 3, 9, 27, \ldots

Example 13

medium

An infinite geometric series has

a_1=8

and

r=\frac{1}{2}

. Find its sum.

Example 14

medium

A culture of bacteria triples every hour, starting at 50. How many after 4 hours?

Example 15

medium

Express the repeating decimal

0.\overline{4}

as a fraction using an infinite geometric series.

Example 16

medium

The 3rd term is 20 and the 5th term is 80 (positive ratio). Find the 1st term.

Example 17

challenge

Three positive numbers form a geometric sequence with product 64 and sum 14. Find them.

Example 18

challenge

For what values of

x

does the infinite geometric series

1+x+x^2+\cdots

converge, and what is its sum?

Example 19

challenge

A geometric sequence has

a_1=3

, and the sum of the first 3 terms is 39. Find the positive ratio

r

Example 20

medium

The 1st term is 81 and the 4th term is 3 (positive ratio). Find

r

Example 21

medium

Find the sum of the first 5 terms of

2, 6, 18, 54, 162

Example 22

medium

An infinite geometric series sums to 12 with first term 9. Find the common ratio.

Example 23

easy

Find the common ratio of the geometric sequence

7, 21, 63, 189, \ldots

Example 24

easy

A geometric sequence has

a_1 = 1

and

r = \tfrac{1}{2}

. Write the first five terms.

Example 25

easy

For

a_n = 3 \cdot 4^{n-1}

, find

a_4

Example 26

medium

A geometric sequence has

a_3 = 12

and

a_6 = 324

. Find

r

Example 27

medium

A geometric sequence has

a_1 = 81

and

r = \tfrac{1}{3}

. Find

a_5

Example 28

medium

Find

S_5

for a geometric sequence with

a_1 = 2

and

r = -2

Example 29

medium

A car worth $24{,}000 depreciates by

15\%

each year. What is its value after 5 years (to the nearest dollar)?

Example 30

medium

In a geometric sequence,

a_4 = 54

and

a_7 = 1458

. Find

a_1

Example 31

medium

Which term of the sequence

3, 6, 12, 24, \ldots

equals

1536

Example 32

hard

The sum of the infinite geometric series

a + ar + ar^{2} + \cdots

20

, and

a = 5

. Find

r

Example 33

hard

Three consecutive terms of a geometric sequence are

x - 2

x + 2

5x - 2

. Find all possible

x

Example 34

hard

An investment of $2000 earns

6\%

compounded annually. After how many full years does it first exceed $5000?

Example 35

hard

Find

r

S_{\infty} = 12

and

a_1 = 3

Example 36

hard

In a geometric sequence with positive terms,

a_2 + a_4 = 30

and

a_3 + a_5 = 60

. Find

r

Example 37

challenge

The sum of the first three terms of a geometric sequence is

14

and their product is

64

. Find the terms (assume positive ratio).

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Related Concepts

Sequence Exponents Series Exponential Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

sequenceexponents