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

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

Answer: Option D
:
D

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

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

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

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

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

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