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Let $a_n$ be the Fibonacci sequence defined by

$$a_{n+1} = a_n + a_{n-1} \ \ \text{with} \ \ a_0 = a_1 = 1 $$

Let $$r_n = \frac{a_n}{a_{n+1}}$$

I have already found two subsequences which are $\{r_{2k-1}\}$ bounded above by $2$ and increasing (thus is convergent), and $\{r_{2k}\}$, bounded below by $1$ and decreasing (thus is convergent,too)

Besides, $$ \lim_{k \to \infty} r_{2k-1} = \ell_1 $$ I found that $l_1^2 - l_1 - 1 = 0$ from above equation.

However, i cannot find $l_2$ where $ \lim\limits_{k \to \infty} r_{2k} = \ell_2.$

Michael Hardy
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Furkan Şahin
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1 Answers1

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$$r_{2k}=\frac{a_{2k}}{a_{2k+1}}=\frac{a_{2k}}{a_{2k}+a_{2k-1}}=\frac1{1+r_{2k-1}}.$$

Then

$$l_2=\frac1{1+l_1}.$$