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
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- Proofs Without Words
- Sums of Geometric Series - Proofs Without Words
- Sine of the Sum Formula
- Parallelogram Law: A PWW
- Parallelogram Law
- Ceva's Theorem: Proof Without Words
- Viviani's Theorem
- A Property of Rhombi
- Triangular Numbers in a Square
- PWW: How Geometry Helps Algebra
- Varignon's Theorem, Proof Without Words
Copyright © 1996=2008 Alexander Bogomolny