Question

If $x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{2\cdots \infty }}}}$, then value of $x$, is

• A. $\sqrt{\frac{5}{2}}$
• B. $\sqrt{\frac{3}{2}}$
• C. $\sqrt{\frac{7}{3}}$
• D. $\sqrt{\frac{5}{3}}$

Answer 1

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

$x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{\mathrm{2....}\infty }}}}$

We can see that ,

$x-1=\frac{1}{3+\frac{1}{2+\frac{1}{2}+\frac{1}{\mathrm{2......}\infty }}}$

Substituting it in the given question we get:

$\Rightarrow x=1+\frac{1}{3+\frac{1}{2+x-1}}=\frac{x+1}{3x+3+1}$

$\Rightarrow x=\frac{4x+5}{3x+4}\Rightarrow 3x^2+4x=4x+5$

$\Rightarrow 3x^2=5\Rightarrow x^2=\frac{5}{3}\Rightarrow x=\pm \sqrt{\frac{5}{3}}$

Hence, option 'D' is correct.