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

Setand for $n\ge 0$ defineThen $p_n/q_n=[a_0,\dots ,a_n]$ is the $n$th convergent.

For a variable final tail $x$, induction on $n$, or multiplication of the continued-fraction matrices, givesIndeed, the identity is immediate for $n=0$, and replacing the tail by $a_{n+1}+1/x$ gives the recurrence above. Taking $x=\gamma >0$ proves the formula.

Solved by gpt-5.6-sol high.