N2 = 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 Tn = n(n+1)/2, viz., a sum of two consecutive triangular numbers is a square:
The algebraic derivation is straightforward:
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n(n + 1)/2 + (n - 1)n/2 = n/2·(n + 1 + n - 1) = n/2·2n = n2.
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The applet attempts to present a visual argument, as a proof without words.
Anirudh Deshpande, India, has observed that, by definition,
It follows that the basic identity Tn-1 + Tn = n2 can be written in a seemingly more profound form:
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Tn-1 + Tn = (Tn - Tn - 1)2.
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On the Web
- An online and iPod video by Julio de la Yncera
Copyright © 1996=2008 Alexander Bogomolny
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