Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 11I / c / Solution

Because each matrix $M(a)=\left(\begin{smallmatrix}a&1\\1&0\end{smallmatrix}\right)$ is symmetric, reversing the partial quotients transposes their product. If the convergents of $[a_0,\ldots,a_n]$ are $p_j/q_j$, then the last two convergents of $[a_n,\ldots,a_0]$ are
$$
\frac{p_n}{p_{n-1}}
\quad\hbox{and}\quad
\frac{q_n}{q_{n-1}}.
$$


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 Incoming links: Reversal identity for a finite continued fraction