# How do you tell whether the sequence 2,4,8,16,.... is arithmetic, geometric or neither?

May 1, 2016

Geometric as far as it goes.

#### Explanation:

It is geometric as far as it goes.

Notice that the ratio between each successive pair of terms is constant:

$\frac{4}{2} = 2$

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

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

That these ratios are all the same is sufficient for the given terms to form a geometric sequence with general term:

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

The sequence then continues:

$2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 , 1024 , 2048 , \ldots$

The question is in the "...". Any finite number of terms does not determine an infinite sequence.

For example we can match the sequence $2 , 4 , 8 , 16$ with a cubic polynomial:

${a}_{n} = \frac{1}{3} \left({n}^{3} - 3 {n}^{2} + 8 n\right)$

Then we would find that the next terms would be $30 , 52 , 84 , \ldots$