I'm working on a question from a past exam paper,
Let $C_n=\frac{p_q}{q_n}$ be the convergents of a finite continued fraction. State the $p$-$q$ relation and prove it.
I don't yet have access to course notes. Wikipedia gives a way to calculate $p_{n+1}$ from $p_n$ and $q_{n+1}$ from $q_n$, but I don't see any mention of a $p$-$q$ relation. This answer states that each pair $p_n,q_n$ is coprime and gives the relation $p_nq_{n+1}-q_np_{n+1}=\pm1$. But the $p_q$ notation in the numerator of the exam question suggests that $p$ can be actually expressed in terms of $q$. Can it?