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Math Help - Difference Equation Help

  1. #1
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    Difference Equation Help

    Heres the question:

    Consider the following difference equation
    P(n) = P(n-1) + k*P(n-1)*(A - P(n-1))

    (a) Show that A - P(n) = (1 - kP(n-1))*(A - P(n-1))
    I have already solved a and am just including it incase it is needed to solve b

    (b) If 1-kA>0 and P(n-1)<A, show that P(n)<A
    b is the one I need help with and I really dont know what to do...
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  2. #2
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    Quote Originally Posted by mrtwigx View Post
    Heres the question:

    Consider the following difference equation
    P(n) = P(n-1) + k*P(n-1)*(A - P(n-1))

    (a) Show that A - P(n) = (1 - kP(n-1))*(A - P(n-1))
    I have already solved a and am just including it incase it is needed to solve b

    (b) If 1-kA>0 and P(n-1)<A, show that P(n)<A
    b is the one I need help with and I really dont know what to do...
    NOTE: I'm assuming k is non-negative, although that is not part of your problem statement.

    Since P(n-1) < A and k >= 0,
    kP(n-1) < kA, so
    1 - kP(n-1) > 1 - kA > 0

    and
    A - P(n-1) > 0

    so by (a)

    A - P(n) = (1 - kP(n-1)) * (A - P(n-1)) > 0
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  3. #3
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    Thanks so much for your reply. I always seem to forget about adding new terms to each side of an equation/inequality.
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