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Math Help - Difference equation

  1. #1
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    Difference equation

    Solve the difference equation:
    p_n = p(1-p_{n-1})+(1-p)p_{n-1}
    Any help would be greatly appreciated.
    Thanks
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  2. #2
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    Quote Originally Posted by tbyou87 View Post
    Solve the difference equation:
    p_n = p(1-p_{n-1})+(1-p)p_{n-1}
    Any help would be greatly appreciated.
    Thanks
    p is just a constant? Please let me change to a = p, just so there is less chance of confusion.
    p_n = a(1 -p_{n - 1}) + (1 - a)p_{n - 1}

    p_n = a + (1 - 2a)p_{n - 1}

    p_n + (2a - 1)p_{n - 1} = a

    So we also know that
    p_{n + 1} + (2a - 1)p_n = a

    So
    p_{n + 1} + (2a - 1)p_n = a = p_n + (2a - 1)p_{n - 1}

    or
    p_{n + 1} + (2a - 2)p_n + (1 - 2a)p_{n - 1} = 0

    This is now a homogeneous linear difference equation. Can you solve it from here?

    -Dan
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