Question

The value of $4+\frac{1}{5+{\displaystyle \frac{1}{4+{\displaystyle \frac{1}{5+{\displaystyle \frac{1}{4+.........\infty }}}}}}}$ is

• A. $2+\frac{4}{\sqrt{5}}\sqrt{30}$
• B. $4+\frac{4}{\sqrt{5}}\sqrt{30}$
• C. $2+\frac{2}{5}\sqrt{30}$
• D. $5+\frac{2}{5}\sqrt{30}$

Answer 1

The correct option is C

$2+\frac{2}{5}\sqrt{30}$

Explanation for the correct answer:

Let $y=4+\frac{1}{5+{\displaystyle \frac{1}{4+{\displaystyle \frac{1}{5+{\displaystyle \frac{1}{4+.........\infty }}}}}}}$

$\Rightarrow$ $y=4+\frac{1}{5+{\displaystyle \frac{1}{y}}}$

$\Rightarrow$ $y=4+\frac{y}{5y+{\displaystyle 1}}$

$\Rightarrow$ $y=\frac{20y+4+y}{5y+{\displaystyle 1}}$

$\Rightarrow$ $5y^2+y=21y+4$

$\Rightarrow$ $5y^2-20y-4=0$

$\Rightarrow$ $y=\frac{20\pm \sqrt{{20}^2-4\times \left(-4\right)\times 5}}{2\times 5}$ $\left(\because y=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\right)$

$\Rightarrow$ $y=\frac{20\pm \sqrt{400+80}}{10}$

$\Rightarrow$ $y=\frac{20\pm 4\sqrt{30}}{10}$

$\Rightarrow$ $y=2\pm \frac{2}{5}\sqrt{30}$

Hence, the value of the given expression is $2+\frac{2}{5}\sqrt{30}$.

Hence, option $\left(C\right)$ is the correct answer.