Question

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

• A. \( \frac{\sqrt{11} + 3}{2} \)
• B. \( \frac{\sqrt{11} - 3}{2} \)
• C. \( \frac{\sqrt{15} + 3}{2} \)
• D. \( \frac{\sqrt{15} - 3}{2} \)

Hint:

In continued fraction problems, assume the entire fraction repeats and solve the resulting quadratic equation.

Answer 1

The Correct Option is B

This is a continued fraction problem. By solving the recurrence relation or simplifying the continued fraction, we arrive at the solution \( y = \frac{\sqrt{11} - 3}{2} \). \[ \boxed{\frac{\sqrt{11} - 3}{2}} \]