A geometric sequence is a set of ordered numbers for which the ratio of successive terms is the same.

What is a Geometric Sequence?

In a geometric sequence, the first term is multiplied by a number called the common ratio to create the second term which is multiplied by the common ratio to create the third term and so on.

The common ratio, or $r$, is calculated as follows:
$r=\dfrac {t_2}{t_1}$

Like in arithmetic sequences, the first term is denoted by $a$.

To find whether a sequence is geometric, all terms must be tested for a common ratio. This is calculated as follows:
$r=\dfrac {t_2}{t_1} =\dfrac {t_3}{t_2} = \dfrac {t_4}{t_3} =...$

Example 1

Which of the following are geometric sequences?

a) $4, 16, 64, 256, ...$

$\dfrac {t_2}{t_1}=\dfrac {16}{4}$
$\dfrac {t_2}{t_1}=4$

$\dfrac {t_3}{t_2}=\dfrac {64}{16}$
$\dfrac {t_3}{t_2}=4$

$\dfrac {t_4}{t_3}=\dfrac {256}{64}$
$\dfrac {t_4}{t_3}=4$

Check that the ratio is the same.

There is a common ratio of 4. Therefore this is a geometric sequence where $a=4$ and $r=4$.

b) $-5, 10, -20, 40, ...$

$\dfrac {t_2}{t_1}=\dfrac {10}{-5}$
$\dfrac {t_2}{t_1}=-2$

$\dfrac {t_3}{t_2}$ $\dfrac {-20}{10}$
$\dfrac {t_3}{t_2}=-2$

$\dfrac {t_4}{t_3}=\dfrac {40}{-20}$
$\dfrac {t_4}{t_3}=-2$

Check that there is a common ratio.

There is a common ratio of -2. Therefore this is a geometric sequence where $a=-5$ and $r=-2$.

c) $8, -24, -72, 216$

$\dfrac {t_2}{t_1}=\dfrac {-24}{8}$
$\dfrac {t_2}{t_1}=-3$

$\dfrac {t_3}{t_2}=$ $\dfrac {-72}{-24}$
$\dfrac {t_3}{t_2}=3$

There is not a common ratio so this is not a geometric sequence.