Closed Form for Triangular Numbers/Direct Proof

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

Proof
We have that $\displaystyle \sum_{i=1}^n i = 1 + 2 + \cdots + n$.

Consider $\displaystyle 2 \sum_{i=1}^n i$. Then:

So:

This is the method employed by Gauss who, when very young (according to the apocryphal story), calculated the sum of the numbers from $1$ to $100$ before the teacher had barely sat back down after setting the assignment.