Question Number 219003 by hardmath last updated on 17/Apr/25
Answered by maths2 last updated on 18/Apr/25
$${S}_{{p}} =\Sigma\Sigma\underset{{k}=\mathrm{0}} {\sum}\frac{{k}^{\mathrm{2}} }{\left({n}−{k}\right)^{\mathrm{2}} +{k}^{\mathrm{2}} }=\Sigma\Sigma\underset{{k}=\mathrm{0}} {\sum}\frac{\left({n}−{k}\right)^{\mathrm{2}} }{\left({n}−{k}\right)^{\mathrm{2}} +{k}^{\mathrm{2}} } \\ $$$$\mathrm{2}{S}_{{p}} =\Sigma\Sigma\Sigma\frac{{k}^{\mathrm{2}} +\left({n}−{k}\right)^{\mathrm{2}} }{\left({n}−{k}\right)^{\mathrm{2}} +{k}^{\mathrm{2}} }=\Sigma\Sigma\Sigma\mathrm{1} \\ $$$$=\Sigma\Sigma\left({n}+\mathrm{1}\right)…..{aEasy}\:{from}\:{here} \\ $$