Closed Form for Triangular Numbers/Proof by Polygonal Numbers

Theorem
The closed-form expression for the $n$th triangular number is:
 * $\displaystyle T_n = \sum_{i \mathop = 1}^n i = \frac {n \left({n+1}\right)} 2$

Proof
Triangular numbers are $k$-gonal numbers where $k = 3$.

Hence from Closed Form for Polygonal Numbers we have that:
 * $\displaystyle T_n = \frac {n \left({2 + \left({n-1}\right)\left({k-2}\right)}\right)} 2$

where $k = 3$.

The result follows after some simple algebra.