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Math Help - How do i find an explicit formula for this recursive linear function?

  1. #1
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    How do i find an explicit formula for this recursive linear function?

    a_n = b+ca_{n-1} for constants b and c
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  2. #2
    MHF Contributor chisigma's Avatar
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    We can start with the general case of a first order linear recursive equation...

    a_{n}= \alpha_{n}\ a_{n-1} + \beta_{n} , a_{0}=A (1)

    With a little of patience You find that the solution of (1) is...

    a_{n}= p_{n} (A + \frac{\beta_{0}}{p_{1}} + \frac{\beta_{1}}{p_{2}} + ... + \frac{\beta_{n-1}}{p_{n}}) (2)

    ... where...

    p_{n}= \alpha_{0}\  \alpha_{1}... \alpha_{n-1} (3)

    If \alpha_{n}=b and \beta_{n}= c the (2) becomes...

    a_{n}= b^{n}\ (A + c\ \frac{1-b^{n}}{1-b}) (4)

    Kind regards

    \chi \sigma
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