Question

The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\cdots \infty }}}}$ is:

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

Answer 1

The correct option is B $2+\frac{2}{5}\sqrt{30}$

Let $y=4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\cdots \infty }}}}$
$y=4+\frac{1}{5+\frac{1}{y}}\Rightarrow y=4+\frac{y}{5y+1}\Rightarrow 5y^2-20y-4=0\Rightarrow y=\frac{20\pm \sqrt{400+80}}{10}\Rightarrow y=\frac{20\pm 4\sqrt{30}}{10}\Rightarrow y=\frac{10+2\sqrt{30}}{5}(\because y>0)\therefore y=2+\frac{2}{5}\sqrt{30}$