Sum of Triangular Numbers II (visual proof via cannonballs)

This is a short, animated visual proof showing how a formula for the sum of the first n triangular numbers (which themselves are sums of the first n integers) using a three dimensional stack of spheres (that look like cannonballs). #manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #triangularnumbers #sums #pww​ #proofwithoutwords​ ​ #proof​ #algebra #finitesums #mathematics​ #mathvideo​ #mtbos

Here is an alternate video for a related sum:

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Attribution 4.0 International (CC BY 4.0)

[avyakthaachar.]

Thanks for all your amazing visual proofs, coupled with mesmerizing background audio!

[kedrjack]

It's really cool and beautiful, I never thought of a tetrahedron in that way.

[davranbekrozmetov]

Awesome visuals. Kinda gives me the vibe of the start comparison video I've seen a long ago.
I hope your channel catches up. You are doing great job

[andrewdjang]

I love this one—discovered it and some similar results the other day trying to prove sum of squares (it ultimately worked!). Do you know of any intuitive visual way to illustrate / explain the identity between triangular, tetrahedral, and, in general, simplex numbers, and the multichoose coefficients? Trying to understand that so I can do sums of powers by turning them into sums of simplices. Thanks for all your brilliant and beautiful work!

[comicrelief]

OK, you have derived an equation, then expressed it in a nice sigma notation. Can one go a little further? For instance, what if n is 1000? Can one find or approximate the number of balls without having to use a calculator to solve the series?

[nothummmmmmmus]

I found a formula for any dimensional triangle whit any side