Q:

How is the Fibonacci sequence used in the story of rabbits?

This is how I have approached.
If a pair of rabbits is placed in an enclosed area, how many rabbits will be born in twelve months there if we assume that every month a pair of rabbits produces another pair who mature the next month, and that rabbits begin to bear young two months after their birth? Assume no rabbit dies. 
The number or adult rabbits in a given month is the total number of rabbits in the previous month.
•   Write An for the number of adult pairs in the nth month.
•   Write Rn for the total numbers of pairs in the nth month 
•   This would equal to An= Rn-1
The number of baby pairs in a given month equals the number of adults in the previous month.
•   Write Bn for the number of baby pairs in the nth month.
•   This would mean that: Bn= An-1 = Rn-2
The total number of rabbits including both the adults and babies in the nth month is the sum of the total pairs of rabbits in the previous two months. 
•   Rn = An + Bn = Rn-1 + Rn-2
Continuing like this we would see that after twelve months, there would be an exact 144 pairs of rabbits.
Are my steps reasonable and easy to understand? Is there another way for simplicity's sake?

A:

Assuming $I$ immature and $M$ mature rabbit pairs, on the next month you have $I^+=M$ new immature pairs, and $M^+=I+M$ mature ones.
With the substitutions $M^+:=F_n,I^+:=M:=F_{n-1},I:=F_{n-2},$
this amounts to
$$F_n=F_{n-1}+F_{n-2}.$$

