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

$$itisgiventhat$$$$\underset{r=1}{\sum ^{20}}[f\left(r\right)-10]=200$$$$and$$$$\underset{r=1}{\sum ^{20}}{[f\left(r\right)-10]}^2=2800$$$$findthevalueof$$$$\underset{r=1}{\sum ^{20}}{\left[f\left(r\right)\right]}^2$$$$$$

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

$$wehave\sum _{r=1}^{20}(f\left(r\right)-10)=200\Rightarrow$$$$\sum _{r=1}^{20}f\left(r\right)-200=200\Rightarrow \sum _{r=1}^{20}f\left(r\right)=400also$$$$\sum _{r=1}^{20}(f^2\left(r\right)-20f\left(r\right)+100)=2800\Rightarrow$$$$\sum _{r=1}^{20}{\left(f\left(r\right)\right)}^2-20\sum _{r=1}^{20}f\left(r\right)+2000=2800\Rightarrow$$$$\sum _{r=1}^{20}{\left(f\left(r\right)\right)}^2=800+20\sum _{r=1}^{20}f\left(r\right)$$$$=800+20\times 400=8800\Rightarrow$$$$\sum _{r=1}^{20}{\left(f\left(r\right)\right)}^2=8800.$$$$$$

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