# How do you determine whether the sequence 1, 2, 4, 8, 16,... is arithmetic and if it is, what is the common difference?

It's not arithmetic. It is geometric with a multiplier of 2.

#### Explanation:

We have a sequence of numbers:

$1 , 2 , 4 , 8 , 16 , \ldots$

Is it arithmetic?

An arithmetic sequence will have the same difference between any two consecutive numbers. Let's see if we have that here:

$\left(\begin{matrix}\text{higher consecutive" & "lower consecutive" & "difference} \\ 2 & 1 & 1 \\ 4 & 2 & 2 \\ 8 & 4 & 4 \\ \vdots & \vdots & \vdots\end{matrix}\right)$

And so no - this is not arithmetic.

What then is it?

Another type of sequence is the geometric sequence where the quotient is always the same. Let's test to see if we have that:

$\left(\begin{matrix}\text{higher consecutive" & "lower consecutive" & "quotient} \\ 2 & 1 & 2 \\ 4 & 2 & 2 \\ 8 & 4 & 2 \\ \vdots & \vdots & \vdots\end{matrix}\right)$

And so it is geometric and the multiplier is 2.