Best sum visual proof?

In this video, we show an animation of perhaps the best and most iconic visual proof of the formula for the sum of the first n positive integers.

To see more about this proof and a generalization, check out this video:

This proof is attributed to the "ancient Greeks" by Roger Nelsen (referencing Martin Gardner) and can be found in Roger Nelsen's first compendium, "Proofs Without Words: Exercises in Visual Thinking:

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To learn more about animating with manim, check out:

[brianconley]

35 years old here. Loved geometry class but struggled hard with proofs. It clicked right now. 16 years later.

[hqTheToaster]

There is also a neat equality you can get with e, a power tower, and integration that points to a binomial distribution. Feel free to find out what it is. Great video.

[fsponj]

I remember when I was a kid I made the same formula: (x/2)(x+1).

[anteconfig]

I remember figuring this out on my own about a year or so ago and I’m amazed by how often I find it being used.
It’s like everywhere.

[bubblesort]

I would say the best visual proof indeed

[Xfady]

Brilliant I didn't see it coming 😮😅

[nelsonsiliezar]

Why text book don’t show it this way. 😅

[tbtelbagmllar]

Which program do you use to make videos? And also, amazing job as always

[tamirerez]

This formula for the sum of natural numbers is attributed to the mathematician Gauss, when he was 9 years old.

Johann Carl Friedrich Gauss.
(April 30, 1777 -
February 23, 1855)
was a German mathematician, physicist and astronomer, one of the greatest mathematicians of all time.

[RandyKing]

what are your thoughts on the notion that all your time and effort yield products that make the original problem and solution seem almost too easy? even with specific examples such as this, the viewer can easily fabricate an induction to carry out a full proof. i thoroughly enjoy this channel, thanks for your work!

[greendsnow]

Oh this. Show this to kids instead of making them memorize formulas

[LanceWilson_RYDER]

Please sir can you make visual proof of (a+b)^3 formula big fan👍🙏

[john]

Pls not this type of music. Tired of leveling sounds down