Closed Form for Triangular Numbers/Proof by Telescoping Sum

Proof
Observe that:

Moreover, we have:
 * $\paren {i + 1}^2 - i^2 = 2 i + 1$

And also:
 * $\paren {n + 1}^2 - 1 = n^2 + 2 n$

Combining these results, we obtain:

This concludes the proof.