Question

If $y=2+\frac{1}{4+\frac{1}{4+\frac{1}{4+\dots \dots \infty }}}$, then

• A. $y=6$
• B. $y=5$
• C. $y=\sqrt{6}$
• D. $y=\sqrt{5}$

Answer 1

The correct option is D $y=\sqrt{5}$

Given that $y=2+\frac{1}{4+\frac{1}{4+..........\infty }}$
$y-2=\frac{1}{4+y-2}$
[ $\because$ The fraction is extended upto $\infty$ ,the fractional part of the denominator is $y-2$ ]
$\Rightarrow y^2-4=1\Rightarrow y^2=5$
$\Rightarrow y=\pm \sqrt{5}$
Since, $y>0$
So, $y=\sqrt{5}$
Option D is correct.