Question

How do I find a formula for `s_n` for the sequence -2, 1, 6, 13, 22,...?

Answer 1

I would suggest you to find the formula for `a_n` first.

The idea is that the differences have been increasing by 2.
-2 to 1 is +3
1 to 6 is +5
6 to 13 is +7

So `a_n` should be quadratic.

One can guess that `a_n = n^2+2n-2`

For `S_n` it's just the sum from `a_0` to `a_n`

Good luck!