Simplify: \(1 + \frac{1}{1+\frac{1}{1+\frac{1}{2}}}\)
Answer: A
We solve this continued fraction from the bottom up.
Step 1: \(1+\frac{1}{2} = \frac{3}{2}\).
Step 2: The expression becomes \(1 + \frac{1}{1+\frac{1}{3/2}} = 1 + \frac{1}{1+\frac{2}{3}}\).
Step 3: \(1+\frac{2}{3} = \frac{5}{3}\).
Step 4: The expression becomes \(1 + \frac{1}{5/3} = 1 + \frac{3}{5}\).
Step 5: \(1 + \frac{3}{5} = \frac{5+3}{5} = \frac{8}{5}\).