Another visual proof by @3blue1brown.pic.twitter.com/ekhzIGhyD5
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Which compares nicely to the story we tell students of how Gauss summed 1 + 2 + 3 + ... + n in elementary school.
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The real proof: S=1+2+3+...+(n-1)+n (n terms) S=n+(n-1)+...+3+2+1 (n terms) Add: 2S=(n+1)+(n-1+2)+...+(n-1+2)+(n+1) (n terms) 2S=n x (n+1) or S=n(n+1)/2 (qed)



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Very elegant thanks!
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i d rather stick the two triangles to form a rectangle of n*(n+1) - • • • - - • • - - - •
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same as
@alimahdi1982 and agreed with both of you.
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That makes so much more sense to me. Thank you! I've even seen this proof before and didn't understand it.
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And also an extension to sums of squares:https://nrich.maths.org/5435
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Another perspective: n = •••... ↓ ... • - - - • - - - 2 x (1+2+ .... +n) • - - - ...etc ↑ ≡ • - - - = n x (n+1) • - - - • - - - = n + n²
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