Question

The value of
$x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{2+.....\infty }}}}$ is

• A. $\sqrt{\frac{5}{3}}$
• B. $-\sqrt{\frac{5}{3}}$
• C. $\sqrt{\frac{3}{2}}$
• D. $-\sqrt{\frac{3}{2}}$

Answer 1

The correct option is A $\sqrt{\frac{5}{3}}$

$x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{2+.....\infty }}}}\Rightarrow x-1=\frac{1}{3+\frac{1}{2+x-1}}\Rightarrow x-1=\frac{1}{3+\frac{1}{x+1}}\Rightarrow x-1=\frac{x+1}{3x+3+1}\Rightarrow (x-1)(3x+4)=x+1\Rightarrow 3x^2+x-4=x+1\Rightarrow 3x^2=5\Rightarrow x=\pm \sqrt{\frac{5}{3}}$
As $x>1$
Therefore, $x=\sqrt{\frac{5}{3}}$