Lety =12+13+12+13+....... What is the value of y?

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

Options:

A . -2

B . 2

C .

\frac{(- \sqrt{15} - 3)}{2}

D .

\frac{(\sqrt{15} - 3)}{2}

Answer: Option D
:
D

Ans: (d)

Given:y=

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

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

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

$\frac{(\sqrt{15} - 3)}{2}$ (CAT 2004)

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