# Geometric Sequences

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.

We can write a formula for the $n$ th term of a geometric sequence in the form

${a}_{n}=a{r}^{n}$ ,

where $r$ is the common ratio between successive terms.

Example 1:

$\left\{2,6,18,54,162,486,1458,...\right\}$

is a geometric sequence where each term is $3$ times the previous term.

A formula for the $n$ th term of the sequence is

${a}_{n}=\frac{2}{3}{\left(3\right)}^{n}$

Example 2:

$\left\{12,-6,3,-\frac{3}{2},\frac{3}{4},-\frac{3}{8},\frac{3}{16},...\right\}$

is a geometric series where each term is $-\frac{1}{2}$ times the previous term.

A formula for the $n$ th term of this sequence is

${a}_{n}=24{\left(-\frac{1}{2}\right)}^{n}$

Example 3:

$\left\{1,2,6,24,120,720,5040,...\right\}$

is not a geometric sequence. The first ratio is $\frac{2}{1}=2$ , but the second ratio is $\frac{6}{2}=3$ .

No formula of the form

${a}_{n}=a{r}^{n}$ can be written for this sequence.