Geometric sequences

Use the common ratio to find terms and sums and interpret growth or decay.

Meaning

A geometric sequence forms each next term by multiplying the previous term by a constant common ratio. With first term and ratio the first term is and the th term for is

Example

With first term 81 and ratio the sequence begins 81, 27, 9, 3, …, and its fifth term is 1. The additive change varies, but the multiplicative ratio stays the same.From the first to the fifth term, multiply by the common ratio four times: 81×()⁴=1. Reaching the nth term takes n−1 multiplications, explaining the exponent n−1.A negative common ratio can make signs alternate. You cannot calculate a ratio by division when the preceding term is zero.

Practice for this concept

Scope and references

Topics

Common ratio, general term, geometric mean, sums

Learning goal

Use the common ratio to find terms and sums and interpret growth or decay.

Helpful background

Connect repeated scaling with powers.