Geometric series: sums of powers of 8 (visual proof)

This is a short, animated (wordless) visual proof demonstrating the sums of finite geometric series of powers of 8 using cubes. #math #manim​ #mathvideo​ #calculus #mtbos​​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricsums #series #squares #finitesums #induction #discretemath

Here are two a related proofs for powers of 3, 4 and 9:

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Music in the video:
Creative Commons Attribution 3.0 Unported License

[rajibmukherjee]

I have been following your channel for quite some time. PWWs come alive with your hard work and exquisite presentation. You have presented this work of mine very aptly through animation. I sincerely thank you for presenting the beauty of PWW to us by creating these animations with so much care. Finally, I feel very peaceful when I watch your wonderful works carefully, the background music you add to these animations work as positive catalyst.

[benjaminmathewson]

Brilliant, elegant work. Haven't looked over your collection in too long, so I'm looking forward to getting back to bingeing on the MVP's!!

[MathNerd]

I really like how this exact same visual proof can be generalized to any perfect cube, not just 8. Just replace 8 with k³ and 7 with k³ - 1. Nice video! :)

[snatureblogger]

Congratulations Rajib Mukharjee Sir...❤️
Feeling proud to be your student, .☺️

[isaacewing]

that was awesome!!! i love summation equations

[bikashchakraborty]

Wao! I know Dr. Rajib Mukherjee. He is doing many PWW's in recent time!

[bilashghosh]

Amazing visualization. I am also a student of Rajib Sir. Very glad to see his work👏. Also, hats off to you for your beautiful presentation👌.

[jakobthomsen]

...and this type of visualization should work for hypercubes of all dimensions 🙂

[sagarnayak]

So Can i try to give u some visual proofs