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Math Help - recurrence relations

  1. #1
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    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 r_{n} be the number of pairs of rabbits alive at the end of each month n for each integer n \ge 1 find a recurrence relation for 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 r_{n} = a+4a where a = number of adults, for n \ge 1

    is that correct?

    and part b) would be 41,015,625 pairs so 82,031,250 rabbits
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  2. #2
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    Re: recurrence relations

    oh crap, I meant to post this in discrete

    sorry about that, haven't had a day off in a month
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  3. #3
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    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 |
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