Closed Form for Triangular Numbers

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

Plainly stated: the sum of the first $n$ natural numbers is equal to $\displaystyle \frac {n \left({n+1}\right)} 2$.

This formula pops up frequently in fields as differing as calculus and computer science, and it is elegant in its simplicity.

Also see

 * Compare Integral of Power for $n = 1$:
 * $\displaystyle \int_0^b x \ \mathrm d x = \frac {b^2} 2$

Historical Note
This theorem was proved by during the course of his proofs of the volumes of various solids of revolution in his On Conoids and Spheroids.