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Question Number 33152 by Rio Mike last updated on 11/Apr/18

it is given that  (1/n)Σ_(r=1) ^n  x^r =2 and (√((1/n)Σ_(r=1) ^n (x_r )^2 −(1/n^2 )(Σ_(r=1) ^n )^2 ))= 3  determine in terms of n the value  of.  Σ_(r=1) ^n (x_r +1)^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}^{{r}} =\mathrm{2}\:{and}\:\sqrt{\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} \right)^{\mathrm{2}} −\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\right)^{\mathrm{2}} }=\:\mathrm{3} \\ $$$${determine}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value} \\ $$$${of}. \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} +\mathrm{1}\right)^{\mathrm{2}} \\ $$

Commented by prof Abdo imad last updated on 12/Apr/18

we have  Σ_(r=1) ^n  x^r  =2n  ,(1/n) Σ_(r=1) ^n  x_r ^2   −1=9  because (Σ 1)^2  =n^2  ⇒  Σ_(r=1) ^n x_r ^2   =10n so  Σ_(r=1) ^n  (x_r  +1)^2  = Σ_(r=1) ^n  ( x_r ^2  +2x_r  +1)  = Σ_(r=1) ^n   x_r ^2   + 2 Σ_(r=1) ^n  x_r   + n  =10n +4n  +n  =15n  (  if there is no mistake in  the question)

$${we}\:{have}\:\:\sum_{{r}=\mathrm{1}} ^{{n}} \:{x}^{{r}} \:=\mathrm{2}{n}\:\:,\frac{\mathrm{1}}{{n}}\:\sum_{{r}=\mathrm{1}} ^{{n}} \:{x}_{{r}} ^{\mathrm{2}} \:\:−\mathrm{1}=\mathrm{9} \\ $$$${because}\:\left(\Sigma\:\mathrm{1}\right)^{\mathrm{2}} \:={n}^{\mathrm{2}} \:\Rightarrow\:\:\sum_{{r}=\mathrm{1}} ^{{n}} {x}_{{r}} ^{\mathrm{2}} \:\:=\mathrm{10}{n}\:{so} \\ $$$$\sum_{{r}=\mathrm{1}} ^{{n}} \:\left({x}_{{r}} \:+\mathrm{1}\right)^{\mathrm{2}} \:=\:\sum_{{r}=\mathrm{1}} ^{{n}} \:\left(\:{x}_{{r}} ^{\mathrm{2}} \:+\mathrm{2}{x}_{{r}} \:+\mathrm{1}\right) \\ $$$$=\:\sum_{{r}=\mathrm{1}} ^{{n}} \:\:{x}_{{r}} ^{\mathrm{2}} \:\:+\:\mathrm{2}\:\sum_{{r}=\mathrm{1}} ^{{n}} \:{x}_{{r}} \:\:+\:{n} \\ $$$$=\mathrm{10}{n}\:+\mathrm{4}{n}\:\:+{n}\:\:=\mathrm{15}{n}\:\:\left(\:\:{if}\:{there}\:{is}\:{no}\:{mistake}\:{in}\right. \\ $$$$\left.{the}\:{question}\right) \\ $$

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