Question

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

Answer 1

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:

`3/1 = 9/3 != 7/3`

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

`2, 4, 8, 16, 32,...`

(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`