<s>[INST] Define the sequences $x_n$& $y_n$ for $n \geq 1$ using the relations $x_n = x_{n+1} +2y_{n+1}$ & $y_n = y_{n-1} +x_{n-1}$ $?$ Define the sequences $x_n$& $y_n$ for $n \geq 1$ using the relations $x_n = x_{n+1} +2y_{n+1}$  &  $y_n = y_{n-1} +x_{n-1}$ $?$
$:$ 
$n \geq 1$ ,  $x_0,y_0 \epsilon \mathbb{Z}_{+} $
$x_n = x_{n+1} +2y_{n+1}$ 
$y_n = y_{n-1} +x_{n-1}$
I came across this problem while solving some problems on limits of sequences,where in this question it asks to define the sequence. So, I tried out by finding the difference $d_x$ and $d_y$ as follows,
$d_x$=$x_{n+1}-x_n$= $-2y_{n+1}$ 
$d_y$ =$y_n- y_{n-1}$=$x_{n-1}$ 
then defining the sequence using the difference I got,
$x_n = x_0 -(n-1)2y_{n+1}$ 
$y_n = y_0 +(n-1)x_{n-1}$
But I know this is not the way because, after it asks to prove the limit.After quite researching I came to know about the recurrence relations, but I don't know anything about that.  I'm quite confused!!
 [/INST] Hint.
We can arrange it as
$$
\left[
\begin{array}{c}
x_n\\
y_n
\end{array}
\right] = 
\left[
\begin{array}{cc}
-1 & -2\\
1 & 1
\end{array}
\right]\left[
\begin{array}{c}
x_{n-1}\\
y_{n-1}
\end{array}
\right]
$$
NOTE
For $A = \left[
\begin{array}{cc}
-1 & -2\\
1 & 1
\end{array}
\right]$ 
we have
$$
\left[
\begin{array}{c}
x_n\\
y_n
\end{array}
\right] = A^n \left[
\begin{array}{c}
x_0\\
y_0
\end{array}
\right]
$$
The recurrence behavior can be easily depicted by making
$$
A = \left[
\begin{array}{cc}
 -1 & -2 \\
 1 & 1 \\
\end{array}
\right]
$$
$$
A^2 = \left[
\begin{array}{cc}
 -1 & 0 \\
 0 & -1 \\
\end{array}
\right]
$$
$$
A^3 = \left[
\begin{array}{cc}
 1 & 2 \\
 -1 & -1 \\
\end{array}
\right]
$$
$$
A^4 = \left[
\begin{array}{cc}
 1 & 0 \\
 0 & 1 \\
\end{array}
\right]
$$
$$
A^5 = \left[
\begin{array}{cc}
 -1 & -2 \\
 1 & 1 \\
\end{array}
\right]
$$
etc.
Note the periodicity.
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