Software engineer here; I've written a little program which demonstrates that for the following function:
$f(n+1) = f(n)+n+1$
starting with
$f(0)=0$
the following is true:
$\lim_{n\to \infty}\left(f(n)\right) = \frac{n^2}{2}$
and therefore the Big-O notation which best describes the function is $O(n^2)$. I have no idea how to prove it formally, though. Can anyone help?