Check if $(a_n)$ is arithmetic progression with a given $S_n$ We have $(a_n)$ with $S_n = n^2$. Check if $(a_n)$ is an arithmetic progression or not.
NOTATION
$$(a_n)_{n\ge1}\\S_n = a_1 + a_2 + \cdots + a_n, \forall n\in \mathbb{N}^*$$

$$a_{n+1}-a_n=\left(S_{n+1}-S_n\right)-\left(S_n-S_{n-1}\right)=(n+1)^2-n^2-(n^2-(n-1)^2)=$$
$$=2n+1-(2n-1)=2$$
Then yes: it is an A.P.
