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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

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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 $a_n$.

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  • $\begingroup$ So would it be $1+\frac{1}{a_n}& @robjohn $\endgroup$ Commented Jun 12, 2018 at 9:37
  • $\begingroup$ What would be $1+\frac1{a_n}$? $\endgroup$ Commented Jun 12, 2018 at 10:16
  • $\begingroup$ It in terms of a(n+1) and a(n) $\endgroup$ Commented Jun 12, 2018 at 10:24
  • $\begingroup$ $1+\frac1{a_n}$ is the right hand side. $\endgroup$ Commented Jun 12, 2018 at 14:54
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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 $a_{n-1}$.

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  • $\begingroup$ Where would I start doing the derivation? and how do I know when I have found the recursive relation to the sequence of ratios $\endgroup$ Commented Jun 12, 2018 at 9:13

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