Evaluate the continued fraction:

$$x = 3 + \frac{1}{2 + \frac{1}{5 + \frac{1}{3}}}$$

Question

Evaluate the continued fraction:

$$x = 3 + \frac{1}{2 + \frac{1}{5 + \frac{1}{3}}}$$

Accepted Answer

Solution:
Given continued fraction:
$$ x = 3 + \frac{1}{2 + \frac{1}{5 + \frac{1}{3}}} $$
First solve the innermost part:
$$ 5 + \frac{1}{3} = \frac{15 + 1}{3} = \frac{16}{3} $$
So,
$$ \frac{1}{5 + \frac{1}{3}} = \frac{1}{\frac{16}{3}} = \frac{3}{16} $$
Now the denominator becomes:
$$ 2 + \frac{3}{16} = \frac{32 + 3}{16} = \frac{35}{16} $$
So,
$$ \frac{1}{2 + \frac{1}{5 + \frac{1}{3}}} = \frac{1}{\frac{35}{16}} = \frac{16}{35} $$
Now,
$$ x = 3 + \frac{16}{35} $$
$$ x = \frac{105}{35} + \frac{16}{35} $$
$$ x = \frac{121}{35} $$
Therefore,
$$ \boxed{\frac{121}{35}} $$
Final Answer: **(a) $\frac{121}{35}$**

Answer

Answer

Answer