It is well known that [math]1+2+3+\ldots+n=\frac{n\cdot\left(n+1\right)}{2}.[/math] Also well known, but more difficult to prove that [math]1\cdot1+2\cdot2+3\cdot3+\ldots+n\cdot n=\frac{n\cdot\left(n+1\right)\cdot\left(2n+1\right)}{6}.[/math] Usually this latter formula is proven by using induction, but it hides the geometrical background of the right side of the equation.[br]Recently I read a "proof without words" [url=http://www.maa.org/sites/default/files/Siu15722.pdf]explanation[/url] of this important formula, published by Man-Keung Siu from the University of Hong Kong. Finally I decided to create this visualization in GeoGebra's 3D engine programmed by Mathieu Blossier and the GeoGebra Team.[br]The applet is limited to [math]n\le3[/math] for technical reasons. Although, if you download the material, in the desktop version you may want to increase [math]n[/math]. Nevertheless, the proof is still easy to understand even if [math]n=3[/math].

Actually, the proven formula is [math]\frac{n\cdot(n+\frac{1}{2})\cdot(n+1)}{3}[/math] which is equivalent to the well known product.