Geometric series: sums of powers of 9 are Triangular (visual proof)

This is a short, animated visual proof demonstrating how to visualize the sum of the finite geometric series of powers of 9 by showing that such a sum is a triangular number. #mathshorts​ #mathvideo​ #math​ #numbertheory #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #series #triangularnumbers #geometricsum #geometricseries #finitesums #discretemath

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[DitDede]

While watching the proof, I realized I was smiling. Very nice!
BTW, got here from Mathologer 2.

[VietMath]

Did you compose your own music?? Very impressive!

[alvarol.martinez]

So cool! It feels very specific about 9, which is interesting. I wonder how one would rearrange cubes into the corresponding pyramid* and what the number would be... maybe 3³?
*: the one with three perpendicular edges of equal length

[joeeeee]

Sum from 0 to n-1 of 9^i = (9^n-1)/(9-1)

Is it the same for other numbers apart from 9?

[RandomAmbles]

The music was a bit... much - but this was otherwise an excellent visual demonstration!