Find two sequences $\{x_n\}$, $\{y_n\}$

Question

I need tips/hints (do not solve) for this problem, cause I have no clue on how to do it or where to even start.

Find two sequences $\{x_n\}$, $\{y_n\}$, so that

$$x_{n+1}=x_n+3y_n$$

$$y_{n+1}=2x_n+2y_n$$

when $x_0=5$ and $y_0=-5$

Answer 1

Guide:

$$\begin{bmatrix} x_{n} \\ y_{n}\end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 2\end{bmatrix} \begin{bmatrix} x_{n-1} \\ y_{n-1} \end{bmatrix}$$

• Find a relationship between $\begin{bmatrix} x_{n} \\ y_{n}\end{bmatrix}$ and $\begin{bmatrix} x_{n-2} \\ y_{n-2}\end{bmatrix}$.

• Find a relationship between $\begin{bmatrix} x_{n} \\ y_{n}\end{bmatrix}$ and $\begin{bmatrix} x_{0} \\ y_{0}\end{bmatrix}$.

• Raising power of $\begin{bmatrix} 1 & 3 \\ 2 & 2\end{bmatrix}$ might help. Diagonalization might help.

Answer 2

Hint : use a bidimensional sequence $Z_n:=(x_n,y_n)$ and find the matrix $A$ such that $$ Z_{n+1}=AZ_n $$

Answer 3

Alt. hint:  you can "eliminate" one set of variables and derive a recurrence in the other one alone. For example, the first recurrence gives $\,x_{n+1}-x_n=3y_n\,$, then writing the second one for $n+1$ and $n$ then subtracting the two gives:

$$ y_{n+2}-y_{n+1}=2(x_{n+1}-x_{n})+2(y_{n+1}-y_{n})=6y_n+2(y_{n+1}-y_{n}) \\[10px] \iff \quad y_{n+2}=3y_{n+1}+4y_n $$