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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 11G / b / Solution

The continued-fraction algorithm starts with

a_{0} = 3

. Rationalizing the successive remainders gives

\frac{1}{11 - 3} = \frac{11 + 3}{2} = 3 + \frac{1}{11 + 3},

and

11 + 3 = 6 + (11 - 3) .

The same remainder has returned, so the digits

3, 6

repeat. Therefore

11 = [3; \overline{3, 6}] .

Solved by gpt-5.6-sol high.

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