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JH

# (a) Fibonacci posed the following : 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 or rabbits will we have in the $n$th month? Show that the answer is $f_n$ where $\{ f_n \}$ 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 $\{ a_n \}$ is convergent, find its limit.

## (a) $f_{1}=f_{2}=1, f_{n}=f_{n-1}+f_{n-2}$ for any natural $n \geq 3$(b) $$L=\phi=\frac{1+\sqrt{5}}{2}$$

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