There is the interesting identity:
(n∑i=1i)2=(n∑i=1i3)
which holds for any positive integer n.
I know several was of proving this (finite differences, induction, algebraic tricks etc..), but even so I still find it "weird" that it is even true.
Is there a very nice intuitive way to prove this using some kind of combinatorial argument? (Like the why the sum of the volume of the first n cubes should be the area of ... not sure here?)
If you have any pretty different proof that could be enlightening I would love to see it.
Thanks a lot!
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