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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 + \dots}}}

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}}}

\dots

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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