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Question Number 33153 by Rio Mike last updated on 11/Apr/18

it is given that Σ_(r=1 ) ^(20) [f(r)−10]=200 and Σ_(r=1) ^(20) [f(r)−10]^2 =2800 find the value of Σ_(r=1) ^(20) [f(r)]^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\:\:\underset{{r}=\mathrm{1}\:} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]=\mathrm{200} \\ $$$${and} \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]^{\mathrm{2}} =\mathrm{2800} \\ $$$${find}\:{the}\:{value}\:{of} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)\right]^{\mathrm{2}} \\ $$$$ \\ $$

Commented by prof Abdo imad last updated on 11/Apr/18

we have Σ_(r=1) ^(20) (f(r)−10) =200 ⇒ Σ_(r=1) ^(20) f(r) −200=200 ⇒ Σ_(r=1) ^(20) f(r) =400 also Σ_(r=1) ^(20) ( f^2 (r) −20f(r) +100) =2800 ⇒ Σ_(r=1) ^(20) (f(r))^2 −20 Σ_(r=1) ^(20) f(r) +2000 =2800 ⇒ Σ_(r=1) ^(20) (f(r))^2 = 800 +20 Σ_(r=1) ^(20) f(r) =800 +20 ×400 = 8800 ⇒ Σ_(r=1) ^(20) (f(r))^2 =8800.

$${we}\:{have}\:\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \left({f}\left({r}\right)−\mathrm{10}\right)\:=\mathrm{200}\:\Rightarrow \\ $$$$\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:{f}\left({r}\right)\:−\mathrm{200}=\mathrm{200}\:\Rightarrow\:\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:{f}\left({r}\right)\:=\mathrm{400}\:{also} \\ $$$$\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:\left(\:{f}^{\mathrm{2}} \left({r}\right)\:−\mathrm{20}{f}\left({r}\right)\:+\mathrm{100}\right)\:=\mathrm{2800}\:\Rightarrow \\ $$$$\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:\left({f}\left({r}\right)\right)^{\mathrm{2}} \:−\mathrm{20}\:\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} {f}\left({r}\right)\:+\mathrm{2000}\:=\mathrm{2800}\:\Rightarrow \\ $$$$\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:\left({f}\left({r}\right)\right)^{\mathrm{2}} \:=\:\mathrm{800}\:+\mathrm{20}\:\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} {f}\left({r}\right) \\ $$$$=\mathrm{800}\:+\mathrm{20}\:×\mathrm{400}\:=\:\mathrm{8800}\:\Rightarrow \\ $$$$\sum_{{r}=\mathrm{1}} ^{\mathrm{20}} \:\left({f}\left({r}\right)\right)^{\mathrm{2}} \:\:=\mathrm{8800}. \\ $$$$ \\ $$

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