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

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Answer 1 · 2017-03-05T01:04:19

It is not, since it has no common ratio.

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:

$a_n = 2^n-1$

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