I was trying to come up with a formula for the sums of powers of n from 1 to x (ie, x(x+1)/2 for the first power, x(x+1)(2x+1)/6 for the second, etc), and in the process, I found this pretty cool formula:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] x^r = \sum_{n=1}^{x} n^r - (n-1)^r [/tex]

have any of you seen this before? does it have a name? does it have an easy proof?

it gives an easy way to recursively define the formula for the sum I was looking for, which gave me this other interesting result that I'm sure is nothing new, but I just thought it was also cool:

[tex] \sum_{n=1}^{x} n^3 = [ \sum_{n=1}^{x} n]^2[/tex]

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# What is this formula I found?

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