Sum of Consecutive Triangular Numbers is Square

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

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

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

So:

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