Question

Let $y=\frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+.....}}}}$ What is the value of y?

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Given : $y=\frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+........}}}}$

Thus $y=\frac{1}{2+\frac{1}{3+y}}$

$y=\frac{3+y}{6+2y+1};\Rightarrow 2y^2+7y=3+y;$ On solving, $y=\frac{-3\pm \sqrt{15}}{2};$ since the given fraction is positive, the value of y is $\frac{\sqrt{15}-3}{2}$

You can also go from elimination of answer options. Answer should

$<$ 0.5 and close to it. This happens only at option (d)