The value of 4 + 1 5 + 1 4 + 1 5 + 1 4 + … … ∞ is:

Given $y=4+\frac{1}{5+\underset{y}{\underbrace{\frac{1}{4+\frac{1}{5+\frac{1}{4+\dots \dots \infty }}}}}}$

$y=4+\frac{1}{\left(5+\frac{1}{y}\right)}$

$\Rightarrow y-4=\frac{y}{(5y+1)}$

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

$\Rightarrow y=\frac{20\pm \sqrt{{\left(-20\right)}^2-4·5·\left(-4\right)}}{2·5}$

$\Rightarrow y=\frac{20\pm \sqrt{480}}{10}$

$\therefore y=2-\sqrt{\frac{480}{100}}$$\to$rejected

So, $y=2+\sqrt{\frac{480}{100}}=2+\frac{2}{5}\sqrt{30}$

Correct with Option (A).