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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2 \sum_{i=1}^n i\)	\(=\)	\(\displaystyle 2 \left({1 + 2 + \cdots + n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({1 + 2 + \cdots + n}\right) + \left({1 + 2 + \cdots + n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({1 + n}\right) + \left({2 + \left({n-1}\right)}\right) + \cdots + \left({\left({n-1}\right) + 2}\right) + \left({n + 1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by commutativity and associativity
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({n + 1}\right)_1 + \left({n + 1}\right)_2 + \cdots + \left({n + 1}\right)_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle n \left({n + 1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2 \sum_{i=1}^n i\)	\(=\)	\(\displaystyle n \left({n + 1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \sum_{i=1}^n i\)	\(=\)	\(\displaystyle \frac {n \left({n+1}\right)} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Historical Note

This is the method supposedly 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.

Whether this story is actually true or not is the subject of speculation.