A Dozen Proofs: Sum of Integers Formula (visual proofs) #SoME2

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, area-based 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. 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&theorem proof 14:11 Euler&formula proof 16:10 Water flow diagram proof 17:49 Center of mass proof 19:19 Concluding remarks 20:38 Citations #doublecount That paper includes many references, but here are a few more relevant sources for proofs from this video: Joe DeMaio and Joey Tyson, Proof without words: A graph theoretic summation of the Jaime Gaspar, Proof without words: using trapezoids to compute triangular numbers, Tom Edgar, Proof without words: matchstick...