# Closed Form for Triangular Numbers/Proof using Odd Number Theorem

## Theorem

The closed-form expression for the $n$th triangular number is:

$\displaystyle T_n = \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$

## Proof

 $\displaystyle \sum_{j \mathop = 1}^n \left({2j - 1}\right)$ $=$ $\displaystyle n^2$ Odd Number Theorem $\displaystyle \leadsto \ \$ $\displaystyle \sum_{j \mathop = 1}^n \left({2j - 1}\right) + \sum_{j \mathop = 1}^n 1$ $=$ $\displaystyle n^2 + n$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{j \mathop = 1}^n \left({2 j}\right)$ $=$ $\displaystyle n \left({n + 1}\right)$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{j \mathop = 1}^n j$ $=$ $\displaystyle \frac {n \left({n + 1}\right)} 2$

$\blacksquare$