Chapter 11.1, Problem 83E

(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 nth month? Show that the answer is $f_n$, where is the Fibonacci sequence defined in Example 3(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.