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

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

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

Tn-1 + Tn = 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,

Tn - Tn - 1 = n.

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

Tn-1 + Tn = (Tn - Tn - 1)2.

On the Web

  1. An online and iPod video by Julio de la Yncera

Proofs Without Words

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