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Mathematics · Glossary

What is Geometric sequence?

Also known as: common ratio

Definition 13.7 High School Mathematics · Chapter 13 — Sequences: A First Course

A sequence is geometric with common ratio q≠0q \neq 0 if each term is obtained from the previous one by multiplying by qq:

un+1=q unfor all n.u_{n+1} = q\,u_n \quad \text{for all } n.

Equivalently, when no term vanishes: the ratio un+1un\frac{u_{n+1}}{u_n} is constant, equal to qq.

Equal steps versus equal ratios: an arithmetic sequence (u_n+1 = u_n + 0.9, blue) follows a line, a geometric sequence (u_n+1 = 1.2\,u_n, red) follows an exponential curve that eventually outgrows it.
Equal steps versus equal ratios: an arithmetic sequence (un+1=un+0.9u_{n+1} = u_n + 0.9, blue) follows a line, a geometric sequence (un+1=1.2 unu_{n+1} = 1.2\,u_n, red) follows an exponential curve that eventually outgrows it.
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