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:
 * $T_{n-1} = \dfrac {\left({n - 1}\right) n} 2$
 * $ T_n = \dfrac {n \left({n + 1}\right)} 2$

So:

Illustration

 * SumOfConsecutiveTriangleNumbers.png