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

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}{\dots .}}}}} \Rightarrow 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)

2 x + 2 = x^{2} + 2 x

x^{2} = 2

Answered by amin96 last updated on 29/Nov/21

1 + \frac{1}{1 + \frac{2}{x}} = x \Leftrightarrow \frac{1}{1 + \frac{2}{x}} = x - 1 \Leftrightarrow \frac{x}{x + 2} = x - 1

(x + 2) (x - 1) = x \Leftrightarrow x^{2} + x - 2 = x \Leftrightarrow x^{2} = 2

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