Sum of odd integers: a generalization (visual proof)

This short animated proof demonstrates the classic sum of odds visual proof and then shows one way to extend the idea to finding sums in other polygonal arrays. Surprisingly, the natural extension to finding sums of certain entries in a triangular array yields the sequence of squares. We leave with some open exploration, which can be further learned about at the link below. #math​ #sums #finitesums #sumformula #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #mathematics #weight #mathshorts​ #mathvideo​

If you like visual proofs of sums, check out these videos:

To learn more about animating with manim, check out:

[spikypichu]

So cool as usually! Keep up the great work!

[yurisich]

The color coded nested members of the sequence was great. I wish there were about five or six included to get a better sense of that formation occurring on the left hand side of the resulting k**2 by k**2 area.

[ed.puckett]

Thank you once again for a fascinating proof. Unfortunately, I tried to get the mentioned paper, but it is paywalled. I worry for our society :-/ Regardless, thank you for your work.