Discussion

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: \(1 + \frac{3}{5} = \frac{5+3}{5} = \frac{8}{5}\).

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