1 → ∞ 1\rightarrow\infty

Calculus Level 2

x = 1 1 + 1 1 + 1 1 + 1 1 + ⋯ \LARGE {x=\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\cdots } } } } }

Then find the value of x x up to three decimal.


The answer is 0.618.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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1 solution

Md Mehedi Hasan
Nov 4, 2017

x = 1 1 + 1 1 + 1 1 + 1 1 + ⋯ ⇒ x = 1 1 + x ⇒ x + x 2 = 1 ⇒ x 2 + x − 1 = 0 \LARGE {x=\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\cdots } } } } \\ \Rightarrow x=\frac{1}{1+x}\\ \Rightarrow x+x^2=1\\ \Rightarrow x^2+x-1=0}

Solving x = 0.618 x=0.618 and − 1.618 -1.618 . Here x x can't be a negative number. So, x ≠ − 1.618 x\neq -1.618

Finally x = 0.618 x=\boxed{0.618}

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