Answer
$F_n=F_{n-1}+F_{n-2}$, see explanations.
Work Step by Step
Step 1. Initially, there are two new born rabbits, $F_0=2$
Step 2. In the first month, the pair is not ready to produce more, so $F_1=2$
Step 3. In the second month, the first pair will product another pair so that $F_2=F_1+2=F_1+F_0$
Step 4. In the third month, the first pair will produce one more pair, but the new born will not, so that $F_3=F_2+2=F_2+F1$
Step 5. In the fourth month, all rabbit pairs will be productive, so that $F_4=F_3+4=F_3+F_2$ ... ...
Step 6. We can see the pattern that after $n$th month, the population will be $F_n=F_{n-1}+F_{n-2}$
Step 7. Compare the result with the Fibonacci sequence, we can see that they are the same.
