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Past exam of the mathematics course of the University of Cambridge

2025

Paper 3

The continued-fraction algorithm starts with $a_0=3$. Rationalizing the successive remainders gives
$$
\frac1{\sqrt{11}-3}=\frac{\sqrt{11}+3}{2}
=3+\frac1{\sqrt{11}+3},
$$
and
$$
\sqrt{11}+3=6+(\sqrt{11}-3).
$$
The same remainder has returned, so the digits $3,6$ repeat. Therefore
$$
\boxed{\sqrt{11}=[3;\overline{3,6}]}.
$$


Solution

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

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

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

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