Problem 83E

83. (a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the $n$ th month? Show that the answer is $f_{n}$, where $\left\{f_{n}\right\}$ is the Fibonacci sequence defined in Example $3(\mathrm{c})$.

(b) Let $a_{n}=f_{n+1} / f_{n}$ and show that $a_{n-1}=1+1 / a_{n-2}$. Assuming that $\left\{a_{n}\right\}$ is convergent, find its limit.