Geometric Sequences

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.