= Solution
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.
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