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Question Number 160405 by HongKing last updated on 29/Nov/21
1+(1/(1+(2/(1+(1/(1+(2/(1+(1/(....))))))))))   ⇒  x^2  = ?
$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{1}+\frac{\mathrm{1}}{….}}}}}\:\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{2}} \:=\:? \\ $$
Commented by quvonnn last updated on 29/Nov/21
1+(1/(1+(2/x)))=x  (x+2)+x=x(x+2)  2x+2=x^2 +2x  x^2 =2
$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{2}}{{x}}}={x} \\ $$$$\left({x}+\mathrm{2}\right)+{x}={x}\left({x}+\mathrm{2}\right) \\ $$$$\mathrm{2}{x}+\mathrm{2}={x}^{\mathrm{2}} +\mathrm{2}{x} \\ $$$${x}^{\mathrm{2}} =\mathrm{2} \\ $$
Answered by amin96 last updated on 29/Nov/21
1+(1/(1+(2/x)))=x    ⇔   (1/(1+(2/x)))=x−1   ⇔   (x/(x+2))=x−1  (x+2)(x−1)=x   ⇔  x^2 +x−2=x    ⇔x^2 =2
$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}}=\boldsymbol{\mathrm{x}}\:\:\:\:\Leftrightarrow\:\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}}=\boldsymbol{\mathrm{x}}−\mathrm{1}\:\:\:\Leftrightarrow\:\:\:\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}+\mathrm{2}}=\boldsymbol{\mathrm{x}}−\mathrm{1} \\ $$$$\left(\boldsymbol{\mathrm{x}}+\mathrm{2}\right)\left(\boldsymbol{\mathrm{x}}−\mathrm{1}\right)=\boldsymbol{\mathrm{x}}\:\:\:\Leftrightarrow\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}−\mathrm{2}=\boldsymbol{\mathrm{x}}\:\:\:\:\Leftrightarrow\boldsymbol{\mathrm{x}}^{\mathrm{2}} =\mathrm{2} \\ $$

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