1-1-1-1-1-1-1-1-x-x-3-1-2-

Question Number 30206 by .none. last updated on 18/Feb/18

\frac{1}{1 - \frac{1}{1 - \frac{1}{1 - \frac{1}{1 - x}}}}

x = \frac{\sqrt{3} - 1}{2}

Commented by abdo imad last updated on 18/Feb/18

l e t p u t a = 1 - \frac{1}{1 - \frac{1}{1 - x}} \Rightarrow N = \frac{1}{1 - \frac{1}{a}} = \frac{1}{\frac{a - 1}{a}} = \frac{a}{a - 1} b u t

a = 1 - \frac{1}{\frac{x}{x - 1}} = 1 - \frac{x - 1}{x} = \frac{1}{x} \Rightarrow N = \frac{\frac{1}{x}}{\frac{1}{x} - 1} = \frac{1}{x (\frac{1}{x} - 1)} = \frac{1}{1 - x}

w i t h x = \frac{\sqrt{3} - 1}{2} w e g e t N = \frac{1}{1 - \frac{\sqrt{3} - 1}{2}} = \frac{2}{2 - \sqrt{3} + 1}

N = \frac{2}{3 - \sqrt{3}} .

Answered by ajfour last updated on 18/Feb/18

1 - \frac{1}{1 - x} = \frac{x}{x - 1}

\Rightarrow 1 - \frac{1}{1 - \frac{1}{1 - x}} = 1 - \frac{x - 1}{x} = \frac{1}{x}

\Rightarrow \frac{1}{1 - \frac{1}{(1 - \frac{1}{1 - \frac{1}{x}})}} = \frac{1}{1 - x}

= \frac{1}{1 - (\frac{\sqrt{3} - 1}{2})} = \frac{2}{3 - \sqrt{3}} = \frac{2 (3 + \sqrt{3})}{6}

= 1 + \frac{1}{\sqrt{3}} .

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