The sum of the first n odd integers is n². 1+3+5+...+(2n-1)=n². Animated 3D visualization on a tetrahedron.
The sum of the first n odd integers is n². Viewed on a tetrahedron.
Animated GIF
GeoGebra applet
Proof without words
The sum of the first n odd integers is n². [br]1+3+5+...+(2n-1)=n². [center][math]$\sum_{k=1}^n (2k-1) = n^2$[/math][/center][left]By transforming rows with consecutive odd number of dots (1, 3, 5, etc) into a square I realized it looked like a 3D view of a tetrahedron, so I made this visualization. [br]See here : [url=https://www.geogebra.org/m/wbunwkkg]geogebra.org/m/wbunwkkg[/url][/left]
Which led to this smaller version with a better animation
[url=https://www.geogebra.org/m/kfquc2vb]https://www.geogebra.org/m/kfquc2vb[/url]
GIF
[url=https://www.geogebra.org/m/kfquc2vb]https://www.geogebra.org/m/kfquc2vb[/url]