# N^{2} = N(N+1)/2 + (N-1)N/2

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(Click in the applet area.)

The applet demonstrates a property of triangular numbers _{n} = n(n+1)/2,

T_{n-1} + T_{n} = n^{2}.

The algebraic derivation is straightforward:

n(n + 1)/2 + (n - 1)n/2 = n/2·(n + 1 + n - 1) = n/2·2n = n^{2}.

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 _{n-1} + T_{n} = n^{2}

T_{n-1} + T_{n} = (T_{n} - T_{n - 1})^{2}.

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