Matrix and sequences Given the following matrix : $$
     A=   \begin{bmatrix}
        1 & 1  \\
        1 & 0   \\
        \end{bmatrix}
$$ and the sequences $a_n$, $b_n$, $ c_n$, $d_n$ such that $ A^n$=   \begin{bmatrix}
        a_n& b_n \\
       c_n & d_n  \\
        \end{bmatrix}
 I need to find the pair $(x,y)$ such that $a_{n+2}=xa_{n+1}+ya_n$ for every integer $n$.
What do I need to do? Do I have to find a form for the matrix raised to the power $n$? Can I use Hamilton Cayley here? Could someone please explain in detail because I haven't worked with matrices and sequences until now. The problem also asks me the number of values for $n$ such that $b_n=c_n$ and the limit as $n$ goes to infinity of $\frac{a_n}{b_n}$.

Note that
$$\begin{pmatrix} a_{n+1}&b_{n+1}\\c_{n+1}&d_{n+1} \end{pmatrix} =  \begin{pmatrix} a_n&b_n\\c_n&d_n \end{pmatrix} \begin{pmatrix} 1&1\\1&0 \end{pmatrix} = \begin{pmatrix} a_n + b_n & a_n \\ c_n + d_n & c_n \end{pmatrix} $$
and
$$\begin{pmatrix} a_{n+2}&b_{n+2}\\c_{n+2}&d_{n+2} \end{pmatrix} =  \begin{pmatrix} a_{n+1}&b_{n+1}\\c_{n+1}&d_{n+1} \end{pmatrix} \begin{pmatrix} 1&1\\1&0 \end{pmatrix} = \begin{pmatrix} 2a_n + b_n & a_n + b_n \\ 2c_n + d_n & c_n + d_n \end{pmatrix} $$
So, 
$$ a_{n+2} = 2a_n + b_n$$
$$ a_{n+1} = a_n + b_n $$
Can you solve for $x$ and $y$ from here?
