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?