Set
$$
p_{-2}=0,\qquad p_{-1}=1,\qquad
q_{-2}=1,\qquad q_{-1}=0,
$$
and for $n\geq0$ define
$$
p_n=a_np_{n-1}+p_{n-2},
\qquad
q_n=a_nq_{n-1}+q_{n-2}.
$$
Then $p_n/q_n=[a_0,\ldots,a_n]$ is the $n$th convergent.

For a variable final tail $x$, induction on $n$, or multiplication of the continued-fraction matrices, gives
$$
[a_0,\ldots,a_n,x]
=\frac{p_nx+p_{n-1}}{q_nx+q_{n-1}}.
$$
Indeed, 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.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/11g/a/solution