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Question Number 99025 by joki last updated on 18/Jun/20

Commented by bobhans last updated on 18/Jun/20

$$\mathrm{set}2207-\frac{1}{2207-\frac{1}{2207-\frac{1}{2207-...}}}=q$$$$2207-\frac{1}{q}=q\Rightarrow q^2-2207q+1=0$$$$q=\frac{2207\pm \sqrt{4870845}}{2}=\frac{2207\pm 2207}{2}$$$$\mathrm{case}\left(1\right)q=2207\Rightarrow \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-...}}}=\sqrt[8]{2207}\approx 2.618034056$$

Commented by joki last updated on 18/Jun/20

$$thankssir$$

Commented by MJS last updated on 18/Jun/20

$$\sqrt{4870845}\ne 2207\Rightarrow \mathrm{answer}\mathrm{is}\mathrm{only}\mathrm{an}\mathrm{approximation}$$

Commented by bobhans last updated on 18/Jun/20

$$\mathrm{yes}\mathrm{sir}.\mathrm{my}\mathrm{answer}\mathrm{is}\mathrm{approximation}$$

Answered by bemath last updated on 18/Jun/20

$$\mathrm{let}\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207}-...}}=\mathrm{w}$$$${\mathrm{w}}^8=2207-\frac{1}{{\mathrm{w}}^8}\Rightarrow {\mathrm{w}}^{16}-{2207\mathrm{w}}^8+1=0$$

Commented by mr W last updated on 18/Jun/20

$$wrongsir!$$$$shouldbe$$$${\mathrm{w}}^8=2207-\frac{1}{{\mathrm{w}}^8}$$

Commented by joki last updated on 18/Jun/20

$$havedone?whatisthevaluew?$$

Commented by Rasheed.Sindhi last updated on 18/Jun/20

$$Sirbemath$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$Pleasesaysomethingwhether$$$${it}^{\prime }srightorwrong.$$

Commented by bemath last updated on 18/Jun/20

$$\mathrm{sorry}\mathrm{sir}.\mathrm{i}\mathrm{forgot}.\mathrm{yes}\mathrm{sir}\mathrm{your}\mathrm{answer}$$$$\mathrm{is}\mathrm{correct}$$

Commented by Rasheed.Sindhi last updated on 18/Jun/20

$$Thanxsir!$$

Answered by MJS last updated on 18/Jun/20

$$x=2207-\frac{1}{x}$$$$x^2-2207x+1=0$$$$x=\frac{2207\pm 987\sqrt{5}}{2};x\approx 2207\Rightarrow x=\frac{2207+987\sqrt{5}}{2}$$$$\sqrt[8]{x}=\sqrt{\sqrt{\sqrt{x}}}$$$$\sqrt{\frac{2207+987\sqrt{5}}{2}}=\frac{47+21\sqrt{5}}{2}$$$$\sqrt{\frac{47+21\sqrt{5}}{2}}=\frac{7+3\sqrt{5}}{2}$$$$\sqrt{\frac{7+3\sqrt{5}}{2}}=\frac{3+\sqrt{5}}{2}$$$$$$

Commented by mr W last updated on 18/Jun/20

$$x=\frac{2207-987\sqrt{5}}{2}isalsosolution?$$

Commented by MJS last updated on 18/Jun/20

$$\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{polynome},\mathrm{yes}.\mathrm{but}$$$$\frac{2207-987\sqrt{5}}{2}\ne 2207-\frac{1}{2207-\frac{1}{2207...}}$$

Commented by bobhans last updated on 18/Jun/20

$$\mathrm{great}\mathrm{sir}...$$

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