How to approach this question about recursive sequences

Question

Just practicing some questions about sequences and came across the one below

The sequence $(F_n)$ of Fibonacci numbers is defined by the recursive relation

$F_{n+2} = F_{n+1} + F_n\ $ with $F_1=F_2 = 1$

Use the recursive relation for $F_n$ to find a recursive relation for the sequence of ratios

$a_n=\frac{F_{n+1}}{F_n}$

I have never seen anything like this before and was just wondering if someone could give me the starting Idea

Answer 1

Divide the recursive equation by $F_{n+1}$: $$ \frac{F_{n+2}}{F_{n+1}}=1+\frac{F_n}{F_{n+1}} $$ Write that in terms of $a_{n+1}$ and $a_n$.

Answer 2

Since $F_{n+1}=F_n+F_{n-1}$ then

$a_n = \frac{F_n+F_{n-1}}{F_n}$

From this you should be able to derive an expression for $a_n$ in terms of $a_{n-1}$.