How to compute this sum

I’m trying to remove the summation sign from this formula, is this possible?

1+500k=1(4(2k+1)212k)

Answer

Hints: expand 4(2k+1)212k and use

nk=1k=n(n+1)2

nk=1k2=n(n+1)(2n+1)6.

You can find a proof of the second formula in this post of mine (in Portuguese).

Attribution
Source : Link , Question Author : Timo Willemsen , Answer Author : Américo Tavares

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