1. Recursive sequences equation system

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

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

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

Can you please tell me how can I find the sequences using linear algebra?

2. Re: Recursive sequences equation system

Originally Posted by cristi92
Find the sequences $\displaystyle (x_{n})$ , $\displaystyle (y_{n})$ and $\displaystyle (z_{n})$

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

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

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

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

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

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

See Applications of diagonalisation.

3. Re: Recursive sequences equation system

Thank you very much!