Question

Solve:
$\frac{1}{2-\frac{1}{2-\frac{1}{2-x}}},x\ne 2$

Answer 1

We have
$\frac{1}{2-\frac{1}{2-\frac{1}{2-x}}},x\ne 2$

$\Rightarrow x=\frac{1}{2-\frac{1}{\frac{2(2-x)-1}{2-x}}}\Rightarrow x=\frac{1}{2-\frac{2-x}{4-2x-1}}\Rightarrow x=\frac{1}{2-\frac{2-x}{3-2x}}$

$\Rightarrow x=\frac{3-2x}{2(3-2x)-(2-x)}\Rightarrow x=\frac{3-2x}{4-3x}\Rightarrow x(4-3x)=(3-2x)$

$\Rightarrow 4x-3x^2=3-2x\Rightarrow 3x^2-6x+3=0\Rightarrow x^2-2x+1=0\Rightarrow {(x-1)}^2=0\Rightarrow x=1,1$