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Question Number 77741 by BK last updated on 09/Jan/20

Commented by mr W last updated on 09/Jan/20

Σ_(n=1) ^(10) (n^2 /(n^2 −10n+50))=? A_n =(n^2 /((n−5)^2 +5^2 )) A_(5−k) =(((5−k)^2 )/(k^2 +5^2 )) A_(5+k) =(((5+k)^2 )/(k^2 +5^2 )) A_(5−k) +A_(5+k) =(((5−k)^2 +(5+k)^2 )/(k^2 +5^2 ))=2 A_1 +A_2 +A_3 +...+A_(10) =(A_1 +A_9 )+(A_2 +A_8 )+(A_3 +A_7 )+(A_4 +A_6 )+A_5 +A_(10) =2+2+2+2+(5^2 /5^2 )+((10^2 )/(2×5^2 )) =2+2+2+2+1+2 =11

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\frac{{n}^{\mathrm{2}} }{{n}^{\mathrm{2}} −\mathrm{10}{n}+\mathrm{50}}=? \\ $$$${A}_{{n}} =\frac{{n}^{\mathrm{2}} }{\left({n}−\mathrm{5}\right)^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} } \\ $$$${A}_{\mathrm{5}−{k}} =\frac{\left(\mathrm{5}−{k}\right)^{\mathrm{2}} }{{k}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} } \\ $$$${A}_{\mathrm{5}+{k}} =\frac{\left(\mathrm{5}+{k}\right)^{\mathrm{2}} }{{k}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} } \\ $$$${A}_{\mathrm{5}−{k}} +{A}_{\mathrm{5}+{k}} =\frac{\left(\mathrm{5}−{k}\right)^{\mathrm{2}} +\left(\mathrm{5}+{k}\right)^{\mathrm{2}} }{{k}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }=\mathrm{2} \\ $$$${A}_{\mathrm{1}} +{A}_{\mathrm{2}} +{A}_{\mathrm{3}} +...+{A}_{\mathrm{10}} \\ $$$$=\left({A}_{\mathrm{1}} +{A}_{\mathrm{9}} \right)+\left({A}_{\mathrm{2}} +{A}_{\mathrm{8}} \right)+\left({A}_{\mathrm{3}} +{A}_{\mathrm{7}} \right)+\left({A}_{\mathrm{4}} +{A}_{\mathrm{6}} \right)+{A}_{\mathrm{5}} +{A}_{\mathrm{10}} \\ $$$$=\mathrm{2}+\mathrm{2}+\mathrm{2}+\mathrm{2}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{5}^{\mathrm{2}} }+\frac{\mathrm{10}^{\mathrm{2}} }{\mathrm{2}×\mathrm{5}^{\mathrm{2}} } \\ $$$$=\mathrm{2}+\mathrm{2}+\mathrm{2}+\mathrm{2}+\mathrm{1}+\mathrm{2} \\ $$$$=\mathrm{11} \\ $$

Answered by key of knowledge last updated on 09/Jan/20

about g(x)=(x^2 /(x^2 −10x+50)): if ((x+y)/2)=5 & f(x)=x^2 −10x+50 ⇒f(x)=f(y) (x=5 is max in f(x)) and (x^2 /(f(x)))+(y^2 /(f(y)))=((2x^2 −20x+100)/(x^2 −10x−50))=2 in this way: (g(1)+g(8))+(g(2)+g(3))+...+g(5)+g(10)= 4×2+2+1=11(it is answer)

$$\mathrm{about}\:\:\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{50}}: \\ $$$$\mathrm{if}\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{2}}=\mathrm{5}\:\&\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} −\mathrm{10x}+\mathrm{50} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{y}\right)\:\:\:\:\left(\mathrm{x}=\mathrm{5}\:\mathrm{is}\:\mathrm{max}\:\mathrm{in}\:\mathrm{f}\left(\mathrm{x}\right)\right) \\ $$$$\mathrm{and}\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{f}\left(\mathrm{x}\right)}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{f}\left(\mathrm{y}\right)}=\frac{\mathrm{2x}^{\mathrm{2}} −\mathrm{20x}+\mathrm{100}}{\mathrm{x}^{\mathrm{2}} −\mathrm{10x}−\mathrm{50}}=\mathrm{2} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{way}: \\ $$$$\left(\mathrm{g}\left(\mathrm{1}\right)+\mathrm{g}\left(\mathrm{8}\right)\right)+\left(\mathrm{g}\left(\mathrm{2}\right)+\mathrm{g}\left(\mathrm{3}\right)\right)+...+\mathrm{g}\left(\mathrm{5}\right)+\mathrm{g}\left(\mathrm{10}\right)= \\ $$$$\mathrm{4}×\mathrm{2}+\mathrm{2}+\mathrm{1}=\mathrm{11}\left(\mathrm{it}\:\mathrm{is}\:\mathrm{answer}\right) \\ $$$$ \\ $$$$ \\ $$

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