# What is the common difference of 1, 2, 4, 8, 16,...?

May 10, 2016

This sequence has a common ratio ($2$), rather than a common difference. It is a geometric sequence, not an arithmetic one.

#### Explanation:

Arithmetic sequences have a common difference. That is, the difference between any two consecutive terms is constant.

For example:

$5 , 8 , 11 , 14 , 17$

is an arithmetic sequence with common difference $3$.

We note that:

$8 - 5 = 11 - 8 = 14 - 11 = 17 - 14 = 3$

$\textcolor{w h i t e}{}$
Geometric sequences have a common ratio. That is, the ratio between any two consecutive terms is constant.

For example:

$1 , 2 , 4 , 8 , 16$

is a geometric sequence with common ratio $2$.

We note that:

$\frac{2}{1} = \frac{4}{2} = \frac{8}{4} = \frac{16}{8} = 2$

$\textcolor{w h i t e}{}$
The infinite sequence $1 , 2 , 4 , 8 , 16 , \ldots$ is a geometric sequence if it continues in similar fashion in the "...", doubling every step.