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2 days agoThe graphical proof is really nice :)
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Cake day: July 2nd, 2023
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Proof by induction?
1±2±3±...±n =(1+n)*n/2plugging that into the right side of the equation to transform it:
((1+n)*n/2)^2 = (1+n)^2*n^2/4=n^2(n^2+2n+1)/4 = (n^4 + 2n^3 +n^2)/4If this holds for n:
1^3 + 2^3 +3^3 + ... + n^3 = (n^4 + 2n^3 +n^2)/4Then for n+1:
(n^4 + 2n^3 +n^2)/4 + (n+1)^3 =? (1+n + 1)^2*(n+1)^2/4(n^4 + 2n^3 +n^2)/4 + (n+1)^3 =? (n^2+4n + 4)(n^2 +2n + 1)/4(n^4 + 2n^3 +n^2)/4 + (n+1)^3 =? (n^4 + 4n^3 + 4n^2 + 2n^3 + 8n^2 + 8n + n^2 + 4n + 4)/4(n^4 + 2n^3 +n^2)/4 + (n+1)^3 =? (n^4 + 2n^3 + n^2)/4 + (4n^3 + 12n^2 + 12n + 4)/4(n+1)(n^2 +2n + 1) =? n^3 + 3n^2 + 3n + 1n^3 + 2n^2 + n + n^2 + 2n + 1 =? n^3 + 3n^2 + 3n + 1n^3 + 3n^2 + 3n + 1 =? n^3 + 3n^2 + 3n + 1Which is obviously true.
So yes, it holds forever.
My response to “capitalism is terrorism” was “Go move to Cuba then”.
I don’t know if this exists in different cultures, but in Czechia we often respond to stupid things with something equally or more stupid to show we don’t take it seriously.
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