A single pair of rabbits one male and one female is born at

A single pair of rabbits (one male and one female!) is born at the beginning of the year. Assume that the rabbit pairs are not fertile during the first month of life but thereafter give birth to four new male/female pairs at the end of each month and assume that no rabbits die. Let r_n = the number of pairs of rabbits alive at the end of month n, for each integer n greaterthanorequalto 2 (a) If r_0 = 1, find a recurrence relation for r_0, r_1, r_2, ... (b) Use your recurrence relation from (a) to find r_10, the number of rabbit pairs there will be at the end of ten months.

Solution

r₁ = 1 + 4 x 1 = 5

r₂ = 1 + 4 x 2 = 9

r₃ = 1 + 4 x 3 = 13

So, the regression relation for n months is given by:

r_n = 1 + 4n

b) r₁₀ = 1 + 4 x 10 = 41

So, Rabbit pairs at the end of 10 months = 41