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Math Help - Recursive sequences equation system

  1. #1
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    Recursive sequences equation system

    Find the sequences (x_{n}) , (y_{n}) and (z_{n})

    \left\{\begin{matrix}x_{n+1}=4x_{n}+6y_{n} \\ y_{n+1}=-3x_{n+1}-5y_{n} \\ z_{n+1}=-3x_{n}-6y_{n}+z_{n} \end{matrix}\right.

    x_{0}=1 , y_{0}=2 , z_{0}=3 .

    Can you please tell me how can I find the sequences using linear algebra?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Recursive sequences equation system

    Quote Originally Posted by cristi92 View Post
    Find the sequences (x_{n}) , (y_{n}) and (z_{n})

    \left\{\begin{matrix}x_{n+1}=4x_{n}+6y_{n} \\ y_{n+1}=-3x_{n+1}-5y_{n} \\ z_{n+1}=-3x_{n}-6y_{n}+z_{n} \end{matrix}\right.

    x_{0}=1 , y_{0}=2 , z_{0}=3 .

    Can you please tell me how can I find the sequences using linear algebra?
    \mathbf{x}_{n+1}=\left[\begin{array}{ccc}4 & 6 & 0\\-3 & -5 & 0\\-3 & -6 & 1\end{array}\right]\mathbf{x}_n

    \mathbf{x}_{n+1}=\mathbf{A}\mathbf{x}_n, where \mathbf{A}=\left[\begin{array}{ccc}4 & 6 & 0\\-3 & -5 & 0\\-3 & -6 & 1\end{array}\right]

    \mathbf{x}_n=\mathbf{A}^n\mathbf{x}_0

    \mathbf{A}^n can be calculated easily by diagonalizing \mathbf{A}.

    See Applications of diagonalisation.
    Last edited by alexmahone; January 10th 2012 at 06:27 AM.
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  3. #3
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    Re: Recursive sequences equation system

    Thank you very much!
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