Fibonacci numbers/Nature/Rabbits/Exercise

Rabbits are born in the middle of the month, the times of gestation is one month, and they reach sexual maturity at the age of two months. Every litter consists of two rabbits, and they live forever.

We start in month ${}1$ with just one couple, whose age is one month. Let ${}f_{n}$ denote the number of couples of rabbits in the ${}n$-th month, hence ${}f_{1}=1$, ${}f_{2}=1$. Prove, by induction, the recursive formula

${}f_{n+2}=f_{n+1}+f_{n}\,.$

This sequence is called the sequence of the Fibonacci numbers. How many of the ${}f_{n}$ couples are in the ${}n$-th month able to reproduce?

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