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Question Number 7738 by 314159 last updated on 13/Sep/16

Commented by Rasheed Soomro last updated on 13/Sep/16

((248((1/(√2)))^2 +496((1/(√4)))^6 +1984((1/(√8)))^8 +...)/(1+5+9+...393))  ((248((1/2))^1 +496((1/4))^3 +1984((1/8))^4 +...)/(1+5+9+...393))  −−−−−−−−−−−−−−−−  1+5+9+....+393  a=1st term=1 , d=common difference=5−1=4  T_n =nth term=393  T_n =a+(n−1)d=1+(n−1)(4)=393                     =4n−3=393                      n=((396)/4)=99  S=1+5+9+...+393  S=(n/2)[a+l]     , l=last term      =((99)/2)[1+393]=197×99  −−−−−−−−−−−−−−−  ((124((1/2))^0 +124((1/4))^2 +248((1/8))^3 +...)/(197×99))  Continue

$$\frac{\mathrm{248}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} +\mathrm{496}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{4}}}\right)^{\mathrm{6}} +\mathrm{1984}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{8}}}\right)^{\mathrm{8}} +...}{\mathrm{1}+\mathrm{5}+\mathrm{9}+...\mathrm{393}} \\ $$$$\frac{\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{1}} +\mathrm{496}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{3}} +\mathrm{1984}\left(\frac{\mathrm{1}}{\mathrm{8}}\right)^{\mathrm{4}} +...}{\mathrm{1}+\mathrm{5}+\mathrm{9}+...\mathrm{393}} \\ $$$$−−−−−−−−−−−−−−−− \\ $$$$\mathrm{1}+\mathrm{5}+\mathrm{9}+....+\mathrm{393} \\ $$$${a}=\mathrm{1}{st}\:{term}=\mathrm{1}\:,\:{d}={common}\:{difference}=\mathrm{5}−\mathrm{1}=\mathrm{4} \\ $$$${T}_{{n}} ={nth}\:{term}=\mathrm{393} \\ $$$${T}_{{n}} ={a}+\left({n}−\mathrm{1}\right){d}=\mathrm{1}+\left({n}−\mathrm{1}\right)\left(\mathrm{4}\right)=\mathrm{393} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{4}{n}−\mathrm{3}=\mathrm{393} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{n}=\frac{\mathrm{396}}{\mathrm{4}}=\mathrm{99} \\ $$$${S}=\mathrm{1}+\mathrm{5}+\mathrm{9}+...+\mathrm{393} \\ $$$${S}=\frac{{n}}{\mathrm{2}}\left[{a}+{l}\right]\:\:\:\:\:,\:{l}={last}\:{term} \\ $$$$\:\:\:\:=\frac{\mathrm{99}}{\mathrm{2}}\left[\mathrm{1}+\mathrm{393}\right]=\mathrm{197}×\mathrm{99} \\ $$$$−−−−−−−−−−−−−−− \\ $$$$\frac{\mathrm{124}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{0}} +\mathrm{124}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{2}} +\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{8}}\right)^{\mathrm{3}} +...}{\mathrm{197}×\mathrm{99}} \\ $$$${Continue} \\ $$$$ \\ $$

Commented by prakash jain last updated on 13/Sep/16

248((1/(√2)))^2 +496((1/(√4)))^4 +1984((1/(√2)))^8 +...  n^(th)  term =248×(2^n )×((1/(√2^n )))^2^n

$$\mathrm{248}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} +\mathrm{496}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{4}}}\right)^{\mathrm{4}} +\mathrm{1984}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{\mathrm{8}} +... \\ $$$${n}^{{th}} \:{term}\:=\mathrm{248}×\left(\mathrm{2}^{{n}} \right)×\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}^{{n}} }}\right)^{\mathrm{2}^{{n}} } \\ $$

Commented by Rasheed Soomro last updated on 13/Sep/16

That means there′s a mistake in the statement  of the question! That was also my guess.

$${That}\:{means}\:{there}'{s}\:{a}\:{mistake}\:{in}\:{the}\:{statement} \\ $$$${of}\:{the}\:{question}!\:{That}\:{was}\:{also}\:{my}\:{guess}. \\ $$

Commented by sandy_suhendra last updated on 13/Sep/16

248((1/2))+496((1/2))^6 +1984((1/2))^(12) +...  =248((1/2))+248((1/2))^5 +248((1/2))^9 +...  =248((1/2)+(1/2^5 )+(1/2^9 )+...)  a= (1/2)  ;_   r = (1/2^4 ) = (1/(16))  S_∞ = (a/(1−r)) = ((1/2)/(1−(1/(16)))) = (8/(15))  so we have :  ((248×(8/(15)))/(197×99 (from Rasheed′s answer))) = ((449)/(66,206))

$$\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{496}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{6}} +\mathrm{1984}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{12}} +... \\ $$$$=\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{5}} +\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{9}} +... \\ $$$$=\mathrm{248}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{5}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{9}} }+...\right) \\ $$$${a}=\:\frac{\mathrm{1}}{\mathrm{2}}\:\:;_{} \:\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{4}} }\:=\:\frac{\mathrm{1}}{\mathrm{16}} \\ $$$${S}_{\infty} =\:\frac{{a}}{\mathrm{1}−{r}}\:=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{16}}}\:=\:\frac{\mathrm{8}}{\mathrm{15}} \\ $$$${so}\:{we}\:{have}\:: \\ $$$$\frac{\mathrm{248}×\frac{\mathrm{8}}{\mathrm{15}}}{\mathrm{197}×\mathrm{99}\:\left({from}\:{Rasheed}'{s}\:{answer}\right)}\:=\:\frac{\mathrm{449}}{\mathrm{66},\mathrm{206}} \\ $$

Commented by Rasheed Soomro last updated on 14/Sep/16

Nice!

$$\mathcal{N}{ice}!\:\: \\ $$

Commented by prakash jain last updated on 13/Sep/16

Ok. So 2,6,8 are correct in question.

$$\mathrm{Ok}.\:\mathrm{So}\:\mathrm{2},\mathrm{6},\mathrm{8}\:\mathrm{are}\:\mathrm{correct}\:\mathrm{in}\:\mathrm{question}. \\ $$

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