# How do you tell whether the sequence 1, 3, 7, 15, 31,... is geometric?

Mar 5, 2017

It is not, since it has no common ratio.

#### Explanation:

This sequence is not geometric as the ratio between successive pairs of terms is not constant. For example:

$\frac{3}{1} = \frac{9}{3} \ne \frac{7}{3}$

Adding $1$ to each element does result in a geometric sequence:

$2 , 4 , 8 , 16 , 32 , \ldots$

(with common ratio $2$) and hence we can write a formula for the $n$th term of our original sequence:

${a}_{n} = {2}^{n} - 1$