Sum of Consecutive Triangular Numbers is Square

Theorem
The sum of two consecutive triangular numbers is a square number.

Proof
Let $T_{n-1}$ and $T_n$ be two consecutive triangular numbers.

From Closed Form for Triangular Numbers‎, we have:
 * $\displaystyle T_{n-1} = \frac {\left({n-1}\right) n} 2$
 * $\displaystyle T_n = \frac {n \left({n+1}\right)} 2$

So: