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Question Number 160405 by HongKing last updated on 29/Nov/21

$$1+\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\frac{1}{....}}}}}\Rightarrow {\mathrm{x}}^2=?$$

Commented by quvonnn last updated on 29/Nov/21

$$1+\frac{1}{1+\frac{2}{x}}=x$$$$(x+2)+x=x(x+2)$$$$2x+2=x^2+2x$$$$x^2=2$$

Answered by amin96 last updated on 29/Nov/21

$$1+\frac{1}{1+\frac{2}{\mathbf{x}}}=\mathbf{x}\iff \frac{1}{1+\frac{2}{\mathbf{x}}}=\mathbf{x}-1\iff \frac{\mathbf{x}}{\mathbf{x}+2}=\mathbf{x}-1$$$$(\mathbf{x}+2)(\mathbf{x}-1)=\mathbf{x}\iff {\mathbf{x}}^2+\mathbf{x}-2=\mathbf{x}\iff {\mathbf{x}}^2=2$$

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