Recurrent equation system How to solve this recurrent equation system?
$$\begin{cases}a_{n+1}=a_n+2b_n\\
b_{n+1}=2a_n+b_n\end{cases}$$
The possible solutions should be
\begin{cases}a_n=\frac{1}{2}3^n+\frac{1}{2}(-1)^n\\
b_n=\frac{1}{2}3^n-\frac{1}{2}(-1)^n\end{cases}

Assume we know $a_1$ and $b_1$.
First, $a_{n+1}+b_{n+1}=3(a_n+b_n) = \cdots = 3^n(a_1+b_1)$.
And secondly, $a_{n+1}-b_{n+1}=-(a_n - b_n) = \cdots = (-1)^n (a_1 - b_1)$. 
And 
$$ a_{n+1} = \frac{(a_{n+1}+b_{n+1})+(a_{n+1}-b_{n+1})}{2}$$
$$ b_{n+1} = \frac{(a_{n+1}+b_{n+1})-(a_{n+1}-b_{n+1})}{2}$$
Done!
