A sequence $c_n$ is defined by the following recursion $c_{n+1} = c_n + c_{n-1}$ for every $n \geq 1$ and $c_0 = 1, c_1 = 2$.
-Let $a_n = \frac{c_{n+1}}{c_n}$, for every $n\geq 0$ and prove that $a_n = 1 + \frac{1}{a_{n-1}}$ for every $n \geq 1$.
-Calculate the limit of the following sequence $\frac{c_1}{c_0}, \frac{c_2}{c_1}, \frac{c_3}{c_2},...,\frac{c_{n+1}}{c_n}$ using the fact that this sequence is convergent
I tried close to everything to obtain the formula mentioned above but I did not succeed. What I obtained is $a_n=1 + \frac{c_{n-1}}{c_n}$ which is obviously wrong. As for calculating the limit, I don't have a clue what is meant by 'using the fact that this sequence is convergent'. Could anyone please help me out? Thank you in advance.