by DivMit
Last Updated August 14, 2019 05:20 AM

$$S_n = \sum_{r=0}^{n-1} \cos^{-1} ( \frac{n^2+r^2+r}{\sqrt{n^4 +r^4+2r^3+2n^2 r^2+2n^2r+n^2+r^2}})$$ find$S_{100}$

Now the denominator of this expression is to big and confusing.I don't know how can we resolve it, because of such big expression in limited amount of time. It needs to be telescopic of some kind, but I don't know how to make it. Help please

Notice that $n^4 +r^4+2r^3+2n^2 r^2+2n^2r+n^2+r^2 = (n^2 + r^2 + r)^2 + n^2$

Hence

$S_n=$

$\sum_{r=0}^{n-1} \cos^{-1} ( \frac{n^2+r^2+r}{\sqrt{(n^2 + r^2 + r)^2 + n^2}})$

$=\sum_{r=0}^{n-1} \tan^{-1} \frac{n}{n^2+r^2+r} $

$=\sum_{r=0}^{n-1} $

$= $

Updated December 10, 2017 22:20 PM

Updated May 24, 2017 12:20 PM

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