Practice Geometric Sequence in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

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

Example 2

easy

Find the common ratio of the geometric sequence

7, 21, 63, 189, \ldots

Example 3

easy

Write the general term for

2, 6, 18, 54, \ldots

Example 4

easy

With

a_1=4

and

r=-2

, find the first four terms.

Example 5

hard

An investment of $2000 earns

6\%

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

Example 6

challenge

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

Example 7

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).

Example 8

hard

The sum of the infinite geometric series

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

20

, and

a = 5

. Find

r

Example 9

medium

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

Example 10

medium

For

a_n = 100 \cdot (0.5)^{n-1}

, what is the limit of

a_n

n \to \infty

Example 11

medium

A geometric sequence has

a_1 = 81

and

r = \tfrac{1}{3}

. Find

a_5

Example 12

challenge

A geometric sequence has

a_1=3

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

r

Example 13

easy

Is the sequence

5, 10, 15, 20, \ldots

geometric? Justify in one line.

Example 14

challenge

For what values of

x

does the infinite geometric series

1+x+x^2+\cdots

converge, and what is its sum?

Example 15

hard

In a geometric sequence with positive terms,

a_2 + a_4 = 30

and

a_3 + a_5 = 60

. Find

r

Example 16

hard

Three consecutive terms of a geometric sequence are

x - 2

x + 2

5x - 2

. Find all possible

x

Example 17

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 18

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 19

easy

Find

r

and

a_6

for the sequence

3, 6, 12, 24, \ldots

Example 20

hard

For what values of

r

does the infinite geometric series

\sum_{n=0}^{\infty} r^{n}

converge?

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Related Concepts

Sequence Exponents Series Exponential Function