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f-x-x-2-x-2-1-then-f-1-1-f-2-1-f-100-1-f-1-2-f-2-2-f-100-2-f-1-100-f-2-100-f-100-100-

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Question Number 210737 by mathlove last updated on 18/Aug/24
f(x)=(x^2 /(x^2 +1)) then f((1/1))+f((2/1))+.....+f(((100)/1))+f((1/2)) +f((2/2))+...+f(((100)/2))+f((1/(100)))+f((2/(100))) +......+f(((100)/(100)))=?
$${f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{1}}\:\:\:\:{then} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{1}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{1}}\right)+…..+{f}\left(\frac{\mathrm{100}}{\mathrm{1}}\right)+{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$+{f}\left(\frac{\mathrm{2}}{\mathrm{2}}\right)+…+{f}\left(\frac{\mathrm{100}}{\mathrm{2}}\right)+{f}\left(\frac{\mathrm{1}}{\mathrm{100}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{100}}\right) \\ $$$$+……+{f}\left(\frac{\mathrm{100}}{\mathrm{100}}\right)=? \\ $$
Answered by mr W last updated on 18/Aug/24
f(x)=(x^2 /(x^2 +1)) f((1/x))=(1/(x^2 +1)) ⇒f(x)+f((1/x))=1 f((1/1))+f((2/1))+f((3/1))+...+f(((100)/1))+ f((1/2))+f((2/2))+f((3/2))+...+f(((100)/2))+ f((1/3))+f((2/3))+f((3/3))+...+f(((100)/3))+ ...... f((1/(100)))+f((2/(100)))+f((3/(100)))+...+f(((100)/(100))) =100×f(1)+(99+98+97+...+1)×1 =100×(1/2)+((99×100)/2) =5000
$${f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)=\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\Rightarrow{f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)=\mathrm{1} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{1}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{1}}\right)+{f}\left(\frac{\mathrm{3}}{\mathrm{1}}\right)+…+{f}\left(\frac{\mathrm{100}}{\mathrm{1}}\right)+ \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{2}}\right)+{f}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)+…+{f}\left(\frac{\mathrm{100}}{\mathrm{2}}\right)+ \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{3}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{3}}\right)+{f}\left(\frac{\mathrm{3}}{\mathrm{3}}\right)+…+{f}\left(\frac{\mathrm{100}}{\mathrm{3}}\right)+ \\ $$$$…… \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{100}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{100}}\right)+{f}\left(\frac{\mathrm{3}}{\mathrm{100}}\right)+…+{f}\left(\frac{\mathrm{100}}{\mathrm{100}}\right) \\ $$$$=\mathrm{100}×{f}\left(\mathrm{1}\right)+\left(\mathrm{99}+\mathrm{98}+\mathrm{97}+…+\mathrm{1}\right)×\mathrm{1} \\ $$$$=\mathrm{100}×\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{99}×\mathrm{100}}{\mathrm{2}} \\ $$$$=\mathrm{5000} \\ $$
Commented by mathlove last updated on 18/Aug/24
thanks mr W
$${thanks}\:{mr}\:{W} \\ $$

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