Recognition of Geometric Sequences

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.

See also

Finding the Terms of a Geometric Sequence
Sum of a Finite Geometric Sequence