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 rr.

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,…7, 21, 63, 189, \ldots.

Example 3

easy
Write the general term for 2,6,18,54,…2, 6, 18, 54, \ldots.

Example 4

easy
With a1=4a_1=4 and r=−2r=-2, find the first four terms.

Example 5

hard
An investment of $2000 earns 6%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 1414 and their product is 6464. Find the terms (assume positive ratio).

Example 8

hard
The sum of the infinite geometric series a+ar+ar2+⋯a + ar + ar^{2} + \cdots is 2020, and a=5a = 5. Find rr.

Example 9

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

Example 10

medium
For an=100⋅(0.5)n−1a_n = 100 \cdot (0.5)^{n-1}, what is the limit of ana_n as n→∞n \to \infty?

Example 11

medium
A geometric sequence has a1=81a_1 = 81 and r=13r = \tfrac{1}{3}. Find a5a_5.

Example 12

challenge
A geometric sequence has a1=3a_1=3, and the sum of the first 3 terms is 39. Find the positive ratio rr.

Example 13

easy
Is the sequence 5,10,15,20,…5, 10, 15, 20, \ldots geometric? Justify in one line.

Example 14

challenge
For what values of xx does the infinite geometric series 1+x+x2+⋯1+x+x^2+\cdots converge, and what is its sum?

Example 15

hard
In a geometric sequence with positive terms, a2+a4=30a_2 + a_4 = 30 and a3+a5=60a_3 + a_5 = 60. Find rr.

Example 16

hard
Three consecutive terms of a geometric sequence are x−2x - 2, x+2x + 2, 5x−25x - 2. Find all possible xx.

Example 17

medium
A ball is dropped from 8080 cm and bounces to 70%70\% of its previous height each time. How high after the 4th bounce?

Example 18

challenge
A square of side 11 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 rr and a6a_6 for the sequence 3,6,12,24,…3, 6, 12, 24, \ldots

Example 20

hard
For what values of rr does the infinite geometric series ∑n=0∞rn\sum_{n=0}^{\infty} r^{n} converge?
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