Lesson 7: Geometric Sequences and Series

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$, is $r$, the common ratio. $n$ is greater than or equal to two.

Examples

• The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$.

Property

For a geometric sequence with first term $a_1$ and common ratio $r$:

Recursive: $a_n = a_{n-1} \cdot r$ for $n > 1$, with $a_1$ given

Property

The sum, $S_n$, of the first $n$ terms of a geometric sequence is

where $a_1$ is the first term and $r$ is the common ratio, and $r$ is not equal to one.