N² = N(N+1)/2 + (N-1)N/2

(Click in the applet area.)

The applet demonstrates a property of triangular numbers T_n = n(n+1)/2, viz., a sum of two consecutive triangular numbers is a square:

T_n-1 + T_n = n².

The algebraic derivation is straightforward:

n(n + 1)/2 + (n - 1)n/2 = n/2·(n + 1 + n - 1) = n/2·2n = n².

The applet attempts to present a visual argument, as a proof without words.

Anirudh Deshpande, India, has observed that, by definition,

T_n - T_{n - 1} = n.

It follows that the basic identity T_n-1 + T_n = n² can be written in a seemingly more profound form:

T_n-1 + T_n = (T_n - T_{n - 1})².

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N2 = N(N+1)/2 + (N-1)N/2