Question

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

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

Answer 1

Answer: (A)

$y=4+\frac{1}{\left(5+\frac{1}{y}\right)}$
$y-4=\frac{y}{(5y+1)}$
$5y^2-20y-4=0$
$y=\frac{20+\sqrt{480}}{10}$
$y=\frac{20-\sqrt{480}}{10}\to$ rejected
$y=2+\sqrt{\frac{480}{100}}$