1. ## recurrence relations

A single pair (male and female) of rabbits is born at the beginning of the year. Assume the following:

1) Each pair is not fertile for their first month bet thereafter give birth to four new male/female pairs at the end of every month

2) no rabbits die

a) let $\displaystyle r_{n}$ be the number of pairs of rabbits alive at the end of each month n for each integer $\displaystyle n \ge 1$ find a recurrence relation for $\displaystyle r_{0},r_{1},r_{2}......$

b) how many rabbits will there be at the end of the year

Month | Babies (in pairs) | Adults (in pairs) | total Pairs (r_{n})
1 |1 |0 |1
__________________________________________________ _______________
2 |4 |1 |5
__________________________________________________ ______________
3 |20 |5 |25
__________________________________________________ ______________
4 |100 |25 |125
__________________________________________________ _________
5 |400 |125 |525
__________________________________________________ ___________
6 |2100 |525 |2625
__________________________________________________ ___________
7 |10500 |2625 |13125
__________________________________________________ ____________
8 |52500 |13125 |65625
__________________________________________________ _____________
9 |262500 |65625 |328125
__________________________________________________ ______________
10 |1312500 |328125 |1640625
__________________________________________________ _______________
11 |6562500 |1640625 |8203125
__________________________________________________ _______________
12 |328125000 |8203125 |41015625

(sorry for the ugly table)

the recurrence relation seems to be $\displaystyle r_{n} = a+4a$ where a = number of adults, for $\displaystyle n \ge 1$

is that correct?

and part b) would be 41,015,625 pairs so 82,031,250 rabbits

2. ## Re: recurrence relations

oh crap, I meant to post this in discrete

3. ## Re: recurrence relations

I think I misunderstood the question all values are in pairs

| month | Adults | babies | maturing | total |
| ----- | ------ | ------ | --------- | ----- |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 1 | 4 | 0 | 5 |
| 3 | 1 | 4 | 4 | 9 |
| 4 | 5 | 20 | 4 | 29 |
| 5 | 9 | 36 | 20 | 65 |
| 6 | 29 | 116 | 36 | 181 |
| 7 | 65 | 260 | 116 | 441 |
| 8 | 181 | 724 | 260 | 1165 |
| 9 | 441 | 1764 | 724 | 2929 |
| 10 | 1165 | 4660 | 1765 | 7589 |
| 11 | 2929 | 11716 | 4660 | 19305 |
| 12 | 7589 | 30356 | 11716 | 49661 |