# 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:
**

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

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:
**

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

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:
**

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

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.

See also arithmetic sequences .