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

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2 |4 |1 |5

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3 |20 |5 |25

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4 |100 |25 |125

__________________________________________________ _________

5 |400 |125 |525

__________________________________________________ ___________

6 |2100 |525 |2625

__________________________________________________ ___________

7 |10500 |2625 |13125

__________________________________________________ ____________

8 |52500 |13125 |65625

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9 |262500 |65625 |328125

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10 |1312500 |328125 |1640625

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11 |6562500 |1640625 |8203125

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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