# How do you determine whether the sequence 1^2, 2^2, 3^2, 4^2, 5^2,... is arithmetic and if it is, what is the common difference?

Feb 17, 2017

No, it is not arithmetic.

#### Explanation:

This is not arithmetic. Arithmetic would mean that a number, $d$ is added to each term. This would mean that the terms of the sequence would need to be equally spaced apart.

This is not the case here. The difference between ${t}_{1}$ and ${t}_{2}$ is $3$, and the difference between ${t}_{2}$ and ${t}_{3}$ is $5$. In fact, this series can be represented as

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

While an arithmetic sequence would be of the form

${t}_{n} = a + \left(n - 1\right) d$

The sequence above can be modeled graphically as a parabola, while any arithmetic sequence would be a straight line.

Hopefully this helps!