# How do you tell whether the sequence 2, 4, 8, 16, 32 is arithmetic?

Jul 4, 2017

This is not an arithmetic sequence, it is a geometric sequence

#### Explanation:

Let's call the terms of the series

${u}_{1} = 2$

${u}_{2} = 4$

${u}_{3} = 8$

${u}_{4} = 16$

${u}_{5} = 32$

To see if the sequence is arithmetic, we calculate

${u}_{n} - {u}_{n - 1}$

If this is constant, we have an arithmetis sequence

${u}_{5} - {u}_{4} = 32 - 16 = 16$

${u}_{4} - {u}_{3} = 16 - 8 = 8$

${u}_{3} - {u}_{2} = 8 - 4 = 4$

${u}_{2} - {u}_{1} = 4 - 2 = 2$

This is not constant, so this is not an arithmetic sequence.

But, we can check to see if it's a geometric sequence

We calculate ${u}_{n} / {u}_{n - 1}$

${u}_{5} / {u}_{4} = \frac{32}{16} = 2$

${u}_{4} / {u}_{3} = \frac{16}{8} = 2$

${u}_{3} / {u}_{2} = \frac{8}{4} = 2$

${u}_{2} / {u}_{1} = \frac{4}{2} = 2$

So this is a geometric sequence of general term

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