# How do you tell whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither?

Jun 8, 2018

The sequence is arithmetic

#### Explanation:

A sequence ${a}_{n}$ is:

• Arithmetic if the difference between two consecute elements, i.e. ${a}_{n} - {a}_{n - 1}$, is constant.
• Geometric if the ratio between two consecute elements, i.e. ${a}_{n} / {a}_{n - 1}$, is constant.

Let's test both to get the answer.

Difference test
Let's consider the difference between every element and the following one:

• ${a}_{2} - {a}_{1} = 1 - \left(- 1\right) = 1 + 1 = 2$
• ${a}_{3} - {a}_{2} = 3 - 1 = 2$
• ${a}_{4} - {a}_{3} = 5 - 3 = 2$

The difference remains constant (it's always $2$), so we may already conclude that the sequence is arithmetic. Let's do the ratio test nevertheless:

Ratio test

Let's consider the ratio between every element and the following one:

• ${a}_{2} / {a}_{1} = \frac{1}{- 1} = - 1$
• ${a}_{3} / {a}_{2} = \frac{3}{1} = 3$
• ${a}_{4} / {a}_{3} = \frac{5}{3} = 1.666 \ldots$

The ratio is always different, so the sequence is not geometric, as expected.