Question-154791

Question Number 154791 by azadsir last updated on 21/Sep/21

Answered by Ar Brandon last updated on 24/Sep/21

S=9^2 +7^2 +5^2 +3^2 +1^2 +(−1)^2 +(−3)^2 +∙∙∙+(−25)^2 =9^2 +7^2 +5^2 +3^2 +1^2 +1^2 +3^2 +∙∙∙+25^2 =Σ_(n=1) ^5 (2n−1)^2 +Σ_(n=1) ^(13) (2n−1)^2 =Σ_(n=1) ^5 (4n^2 −4n+1)+Σ_(n=1) ^(13) (4n^2 −4n+1) =((2(5×6×11))/3)−2(5×6)+5 +((2(13×14×27))/3)−2(13×14)+13 =220−60+5+3276−364+13 =3441−351=3090

$${S}=\mathrm{9}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} +\left(−\mathrm{1}\right)^{\mathrm{2}} +\left(−\mathrm{3}\right)^{\mathrm{2}} +\centerdot\centerdot\centerdot+\left(−\mathrm{25}\right)^{\mathrm{2}} \\ $$$$\:\:\:=\mathrm{9}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\centerdot\centerdot\centerdot+\mathrm{25}^{\mathrm{2}} \\ $$$$\:\:\:=\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\sum}}\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} +\underset{{n}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\:\:\:=\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\sum}}\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{4}{n}+\mathrm{1}\right)+\underset{{n}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{4}{n}+\mathrm{1}\right) \\ $$$$\:\:\:=\frac{\mathrm{2}\left(\mathrm{5}×\mathrm{6}×\mathrm{11}\right)}{\mathrm{3}}−\mathrm{2}\left(\mathrm{5}×\mathrm{6}\right)+\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{2}\left(\mathrm{13}×\mathrm{14}×\mathrm{27}\right)}{\mathrm{3}}−\mathrm{2}\left(\mathrm{13}×\mathrm{14}\right)+\mathrm{13} \\ $$$$\:\:\:=\mathrm{220}−\mathrm{60}+\mathrm{5}+\mathrm{3276}−\mathrm{364}+\mathrm{13} \\ $$$$\:\:\:=\mathrm{3441}−\mathrm{351}=\mathrm{3090} \\ $$

Commented by Ar Brandon last updated on 24/Sep/21

You′re right. It′s supposed to be Σ_(n=1) ^5 instead of Σ_(n=1) ^4 3009+9^2 =3090

$$\mathrm{You}'\mathrm{re}\:\mathrm{right}.\:\mathrm{It}'\mathrm{s}\:\mathrm{supposed}\:\mathrm{to}\:\mathrm{be} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\sum}}\:\mathrm{instead}\:\mathrm{of}\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{4}} {\sum}}\: \\ $$$$\mathrm{3009}+\mathrm{9}^{\mathrm{2}} =\mathrm{3090} \\ $$

Commented by SLVR last updated on 24/Sep/21

Excuse..me..error in caliculation it is 3090corret ...instead of 3009

$${Excuse}..{me}..{error}\:{in}\:{caliculation} \\ $$$${it}\:{is}\:\mathrm{3090}{corret}\:…{instead}\:{of}\:\mathrm{3009} \\ $$

Commented by SLVR last updated on 24/Sep/21

$${sir}….{donot}\:{mistake}\:{me}… \\ $$

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