Question Number 197455 by mathlove last updated on 18/Sep/23
$${f}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{6}}{\mathrm{2}}+\mathrm{6}−{x} \\ $$$${g}\left({x}\right)=\sqrt{{x}^{\mathrm{99}} +{x}^{\mathrm{98}} +{x}^{\mathrm{97}} +…..+{x}} \\ $$$${f}\left({g}\left(\mathrm{1}\right)\right)+{f}\left({g}\left(\mathrm{2}\right)\right)+………+{f}\left({g}\left(\mathrm{100}\right)\right)=? \\ $$
Answered by hmr last updated on 18/Sep/23
$$ \\ $$$${f}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{6}}{\mathrm{2}}+\mathrm{6}−{x} \\ $$$${f}\left({x}\right)={x}+\mathrm{3}+\mathrm{6}−{x} \\ $$$${f}\left({x}\right)=\mathrm{9}\:\rightarrow\:{f}\left({g}\left({x}\right)\right)=\mathrm{9} \\ $$$$ \\ $$$${f}\left({g}\left(\mathrm{1}\right)\right)\:+\:{f}\left({g}\left(\mathrm{2}\right)\right)\:+\:…\:+\:{f}\left({g}\left(\mathrm{100}\right)\right) \\ $$$$=\:\mathrm{9}\:+\:\mathrm{9}\:+\:…\:+\:\mathrm{9} \\ $$$$=\:\mathrm{100}\:×\:\mathrm{9} \\ $$$$=\:\mathrm{900} \\ $$