x = 2+(2/(2+(2/(2+(2/(2+(2/(2+(2/x))))))))) find x.


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Question Number 206155 by dimentri last updated on 08/Apr/24

$x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{x}}}}}$ $find x .$
	Answered by MM42 last updated on 08/Apr/24

	$2 + \frac{2}{x} = \frac{2 x + 2}{x} = a$ $2 + \frac{2}{a} = 2 + \frac{x}{x + 1} = \frac{3 x + 2}{x + 1} = b$ $2 + \frac{2}{b} = 2 + \frac{2 x + 2}{3 x + 2} = \frac{8 x + 6}{3 x + 2} = c$ $2 + \frac{2}{c} = 2 + \frac{6 x + 4}{8 x + 6} = \frac{22 x + 16}{8 x + 6} = \frac{11 x + 8}{4 x + 3} = d$ $x = 2 + \frac{2}{d} = 2 + \frac{8 x + 6}{11 x + 8} = \frac{30 x + 22}{11 x + 8}$ $\Rightarrow 11 x^{2} + 8 x = 30 x + 22$ $11 x^{2} - 22 x - 22 = 0$ $x = 1 \pm \sqrt{3} ✓$
	Answered by Rasheed.Sindhi last updated on 08/Apr/24

	$x = 2 + \frac{2}{x} \Rightarrow x^{2} - 2 x - 2 = 0$ $x_{> 0} = \frac{2 + \sqrt{4 + 8}}{2} = 1 + \sqrt{3}$
	Commented by cortano21 last updated on 08/Apr/24

	$n o .$ $i t n o t x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + . . .}}}$
	Commented by Frix last updated on 08/Apr/24

	$x = 2 + \frac{2}{x}$ $insert x = 2 + \frac{2}{x} on the rhs \Rightarrow$ $x = 2 + \frac{2}{2 + \frac{2}{x}}$ $insert x = 2 + \frac{2}{x} on the rhs \Rightarrow$ $x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{x}}}$ $. . .$
	Answered by MATHEMATICSAM last updated on 08/Apr/24

	Answered by Rasheed.Sindhi last updated on 08/Apr/24

	$x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{\frac{2 x + 2}{x}}}}}$ $x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{2 x}{2 x + 2}}}}$ $x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{x}{x + 1}}}}$ $x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2 + \frac{3 x + 2}{x + 1}}}}$ $x = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2 x + 2}{3 x + 2}}}$ $x = 2 + \frac{2}{2 + \frac{2}{\frac{8 x + 6}{3 x + 2}}}$ $x = 2 + \frac{2}{2 + \frac{3 x + 2}{4 x + 3}}$ $x = 2 + \frac{2}{\frac{11 x + 8}{4 x + 3}}$ $x = \frac{22 x + 16 + 8 x + 6}{11 x + 8}$ $x = \frac{30 x + 22}{11 x + 8}$ $11 x^{2} + 8 x - 30 x - 22 = 0$ $11 x^{2} - 22 x - 22 = 0$ $x^{2} - 2 x - 2 = 0$ $x = \frac{2 + \sqrt{4 + 8}}{2} = 1 + \sqrt{3}$
	Commented by MM42 last updated on 08/Apr/24

	$y e s . t h a n k s$
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