In this video, we explore the famous formula for the sum of the first n positive integers. In particular, we present twelve proofs of the sum formula using induction, areabased techniques, combinatorial techniques, physical techniques, and by using a couple of deep theorems. All of the proofs except the first are visually inspired or have a visual component. #SoME2 #manim #visualproof

Comment with your favorite of these twelve or let me know if you have a different favorite proof of this fact!

This video is my submission to the "Summer of Math Exposition 2" contest. The key takeaway is that we can gain exposure to many areas of mathematics by "thinking deeply of simple things" as suggested by mathematician Arnold Ross.

0:00 Introduction : Think Deeply About Simple Things
1:08 Proof by induction
2:49 Classic visual proof and "reverse and add"
4:45 Triangle area proof
5:30 Fundamental theorem of calculus proof
7:04 Trapezoid area proof
7:55 Double counting proof
9:15 Bijective proof
10:40 Linear recurrence proof
12:10 Pick's theorem proof
14:11 Euler's formula proof
16:10 Water flow diagram proof
17:49 Center of mass proof
19:19 Concluding remarks
20:38 Citations

#sumformula #sumintegers #integers​ #mathvideo​ #math​ #mtbos​ #animation​ #theorem​ #pww​ #proofwithoutwords​​ #proof​ #iteachmath #mathematics #3b1bsome2 #combinatorialproof #combinatorics #integralcalculus #area #areas #bijection #trapezoid #triangle #physics #moments #weight #centerofmass #waterflow #recurrences #linearrecurrence #gauss #doublecount

This video is based on an expository paper written by Tom Edgar and Enrique Treviño compiling 35 proofs of the fact that 1+2+3+...+n = n(n+1)/2; the paper is linked here: https://bit.ly/2UPRzep

That paper includes many references, but here are a few more relevant sources for proofs from this video:

Ian Richards, Proof without Words: sum of integers, Mathematics Magazine (March 1984 page 104): https://www.maa.org/sites/default/fil....

Joe DeMaio and Joey Tyson, Proof without words: A graph theoretic summation of the
first n integers, The College Mathematics Journal 38 (2007), no. 4, 296. (https://www.jstor.org/stable/27646507)

Jaime Gaspar, Proof without words: using trapezoids to compute triangular numbers,
Math. Mag. 91 (2018), no. 3, 206–207. (https://www.jstor.org/stable/48665541)

Tom Edgar, Proof without words: matchstick triangles, College Math. J. 47 (2016),
no. 3, 207. (https://doi.org/10.4169/college.math....)

Tom Edgar, Proof without words: a recursion for triangular numbers and more, Math. Mag.
90 (2017), no. 2, 124–125. (https://doi.org/10.4169/math.mag.90.2...)

Loren C. Larson, A discrete look at 1 + 2 ++ n, The College Mathematics Journal 16 (1985), no. 5, 369–382 (https://doi.org/10.2307/2686996)

David Treeby, A moment’s thought: centers of mass and combinatorial identities, Math.
Mag. 90 (2017), no. 1, 19–25. (https://doi.org/10.4169/math.mag.90.1.19)

Another visual proof of this fact using eight triangular arrays can be found in this video here:    • Video  

If you enjoyed this video, please like and subscribe. Also feel free to leave a comment noting your favorite of the 12 proofs!

To learn more about animating with manim, check out:
https://manim.community
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Music in this video:
Ambiment The Ambient by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Source: http://incompetech.com/music/royalty...
Artist: http://incompetech.com/