Rabbit poupulation Fibonacci sequence

Question :
A young pair of rabbits (one of each sex) is placed on an island. A pair
of rabbits does not breed until they are 2 months old. After they are
2 months old, each pair of rabbits produces another pair each month.
Find a recurrence relation for the number of rabbits on the island
after n months, assuming that no rabbits ever die.

Solution :
Suppose there are A_n rabbits after n months.
Then number of rabbits born in (n+1)th month is : A_n+1 - A_n.
A_n+2
= 2*number of rabbits in nth month + number of rabbits born in (n+1)th month.
= 2*A_n + A_n+1 - A_n
= A_n + A_n+1
= Fibonacci sequence