Question

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

Answer 1

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-(-1)=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= 1/(-1)=-1`
• `a_3/a_2 = 3/1=3`
• `a_4/a_3 = 5/3=1.666...`

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