Check if $(a_n)$ is arithmetic progression with a given $S_n$

Question

We have $(a_n)$ with $S_n = n^2$. Check if $(a_n)$ is an arithmetic progression or not.

NOTATION

$$(a_n)_{n\ge1}\\S_n = a_1 + a_2 + \cdots + a_n, \forall n\in \mathbb{N}^*$$

Answer 1

$$a_{n+1}-a_n=\left(S_{n+1}-S_n\right)-\left(S_n-S_{n-1}\right)=(n+1)^2-n^2-(n^2-(n-1)^2)=$$

$$=2n+1-(2n-1)=2$$

Then yes: it is an A.P.

Answer 2

If $(a_n)_n$ has an arithmetic progression, you can show that its sum is given by n times the arithmetic sum of the first and last term, i.e.

$$S_n = n \frac{a_1 + a_n}{2} = \frac{n}{2} (2a_1 + (n - 1)d),$$

where $d$ is the common difference (which is a real number independent of $n$). If such a sequence $(a_n)_n$ exists, then

$$2 a_1 + (n - 1)d = 2n.$$

This means that $d = \frac{2n - 2a_1}{n - 1}$, which is independent of $n$ if and only if $a_1 = 1$ (in that case $d = 2$).

So it could be an arithmetic progression with first term $1$ and common difference $2$.

EDIT: Since $S_1 = a_1 = 1$, it is an arithmetic progression. Thanks Timbuc!