How can we show these two relations?

Question

I want to show that $$\frac{p_n}{p_{n-1}}=[a_n, \ a_{n-1}, \ \ldots , \ a_1, \ a_0] \ \ \ (n\geq 1)$$ and that $$p_nq_{n-3}-q_np_{n-3}=(-1)^{n-1}(a_na_{n-1}+1)$$ but I don't really have an idea how to start.

Could you give me a hint?

Answer 1

Hint for the second question!!

Use that $p_{n-3}=p_{n-1}-a_{n-1}p_{n-2}$ and $q_{n-3}=q_{n-1}-a_{n-1}q_{n-2}$.

Doing some math you will get $(p_{n}q_{n-1} - q_{n}p_{n-1}) + a_{n-1} ( q_{n}p_{n-2} - p_{n}q_{n-2} ) $

Use induction for the 2 brackets.