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:
 * $T_{n - 1} = \dfrac {\paren {n - 1} n} 2$
 * $T_n = \dfrac {n \paren {n + 1} } 2$

So: