Question Number 184927 by mathlove last updated on 14/Jan/23
$${s}=\frac{\mathrm{2}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{6}}{\mathrm{11}}+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{3}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{4}} }+\centerdot\centerdot\centerdot\centerdot \\ $$$${s}=? \\ $$
Answered by Rasheed.Sindhi last updated on 14/Jan/23
$${s}=\frac{\mathrm{2}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{6}}{\mathrm{11}}+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{3}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{4}} }+\centerdot\centerdot\centerdot\centerdot \\ $$$$\frac{{s}}{\mathrm{11}}=\frac{\mathrm{2}}{\mathrm{11}^{\mathrm{1}} }+\frac{\mathrm{6}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{3}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{4}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{5}} }+\centerdot\centerdot\centerdot\centerdot \\ $$$${s}−\frac{{s}}{\mathrm{11}}=\frac{\mathrm{2}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{4}}{\mathrm{11}^{\:} }+\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{3}} }+… \\ $$$$\frac{\mathrm{10}{s}}{\mathrm{11}}=\mathrm{2}+\mathrm{4}\left(\frac{\mathrm{1}}{\mathrm{11}^{\:} }+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{3}} }+…\right) \\ $$$$\mathrm{10}{s}=\mathrm{22}+\mathrm{44}\left(\frac{\mathrm{1}/\mathrm{11}}{\mathrm{1}−\mathrm{1}/\mathrm{11}}\right) \\ $$$$\mathrm{10}{s}=\mathrm{22}+\mathrm{44}\left(\frac{\mathrm{1}}{\mathrm{11}}\centerdot\frac{\mathrm{11}}{\mathrm{10}}\right)=\mathrm{22}+\frac{\mathrm{22}}{\mathrm{5}}=\frac{\mathrm{132}}{\mathrm{5}} \\ $$$${s}=\frac{\mathrm{132}}{\mathrm{10}×\mathrm{5}}=\frac{\mathrm{66}}{\mathrm{25}} \\ $$
Commented by mathlove last updated on 14/Jan/23
$${thanks}\:{dear} \\ $$
Answered by Rasheed.Sindhi last updated on 14/Jan/23
$$\mathbb{A}\boldsymbol{\mathrm{n}}\mathbb{O}\boldsymbol{\mathrm{ther}}\:\mathbb{W}\boldsymbol{\mathrm{ay}}… \\ $$$$\left({i}\right)\bullet{s}=\frac{\mathrm{2}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{6}}{\mathrm{11}}+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{3}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{4}} }+\centerdot\centerdot\centerdot\centerdot \\ $$$${s}−\mathrm{2}=\frac{\mathrm{1}}{\mathrm{11}}\left(\frac{\mathrm{6}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{1}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{3}} }+\centerdot\centerdot\centerdot\centerdot\right. \\ $$$$\left({ii}\right)\bullet\mathrm{11}\left({s}−\mathrm{2}\right)=\frac{\mathrm{6}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{10}}{\mathrm{11}^{\mathrm{1}} }+\frac{\mathrm{14}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{18}}{\mathrm{11}^{\mathrm{3}} }+\centerdot\centerdot\centerdot\centerdot \\ $$$$\left({ii}\right)−\left({i}\right): \\ $$$$\mathrm{11}\left({s}−\mathrm{2}\right)−{s}=\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{0}} }+\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{1}} }+\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{4}}{\mathrm{11}^{\mathrm{3}} }+… \\ $$$$\frac{\mathrm{10}{s}−\mathrm{22}}{\mathrm{4}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{11}}+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{3}} }+… \\ $$$$\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{11}}}=\frac{\mathrm{11}}{\mathrm{10}} \\ $$$$\mathrm{10}{s}−\mathrm{22}=\frac{\mathrm{44}}{\mathrm{10}} \\ $$$$\mathrm{10}{s}=\frac{\mathrm{22}}{\mathrm{5}}+\mathrm{22}=\frac{\mathrm{132}}{\mathrm{5}} \\ $$$${s}=\frac{\mathrm{132}}{\mathrm{50}}=\frac{\mathrm{66}}{\mathrm{25}} \\ $$