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-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 = (Tn - Tn - 1)2.
On the Web
- An online and iPod video by Julio de la Yncera
Copyright © 1996=2008 Alexander Bogomolny