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Common Ratio of a Geometric Sequence

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Example: Determining if a Sequence Is Geometric

Determine whether each sequence is geometric and, if so, state the common ratio.

ⓐ $4, 8, 16, 32, 64, 128, \dots$

Divide consecutive terms: $\frac{8}{4} = 2$ , $\frac{16}{8} = 2$ , $\frac{32}{16} = 2$ , $\frac{64}{32} = 2$ , $\frac{128}{64} = 2$ . Every ratio equals $2$ , so the sequence is geometric with $r = 2$ .

ⓑ $-2, 6, -12, 36, -72, 216, \dots$

Divide consecutive terms: $\frac{6}{-2} = -3$ , $\frac{-12}{6} = -2$ , $\frac{36}{-12} = -3$ , $\frac{-72}{36} = -2$ , $\frac{216}{-72} = -3$ . The ratios alternate between $-3$ and $-2$ , so the sequence is not geometric and has no common ratio.

Divide consecutive terms: $\frac{9}{27} = \frac{1}{3}$ , $\frac{3}{9} = \frac{1}{3}$ , $\frac{1}{3} = \frac{1}{3}$ , $\frac{\frac{1}{3}}{1} = \frac{1}{3}$ , $\frac{\frac{1}{9}}{\frac{1}{3}} = \frac{1}{3}$ . Every ratio equals $\frac{1}{3}$ , so the sequence is geometric with $r = \frac{1}{3}$ .

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Updated 2026-05-26

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OpenStax Intermediate Algebra

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