Monday, 8 April 2019

combinatorics - Combinatorial reasoning for the identity left(sumni=1iright)2=left(sumni=1i3right)








There is the interesting identity:



(ni=1i)2=(ni=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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