Closed Form for Triangular Numbers/Proof using Binomial Coefficients

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
From Properties of Binomial Coefficients: Particular Values, we have that:
 * $\displaystyle \forall k \in \Z, k > 0: \binom k 1 = k$

Thus: