<s>[INST] System of Recurrence Relations Solve the following System of Recurrence Relation:
$$a_n = 2a_{n-1} - b_{n-1} + 2, a_0 = 0$$
$$b_n = -a_{n-1} + 2b_{n-1} - 1, b_0 = 1$$
Workings:
$b_n - 2b_{n-1} = -a_{n-1} - 1$
$a_n = 2a_{n-1} - b_{n-1} + 2$
$a_{n+1} = 2a_n - b_n + 2$
$-2a_n = -4a_{n-1} + 2b_{n-1} - 4$
$a_{n+1} + 2a_n = (2a_n - b_n + 2) + (4a_{n-1} - 2b_{n-1} + 4) $
$a_{n+1} + 2a_n = 2a_n  + 4a_{n+1} - b_n -2b_{n-1} + 6$
$a_{n+1} + 2a_n = 2a_n + 4a_{n-1} - (-a_{n-1} - 1) + 6$
$a_{n+1} + 2a_n = 2a_n + 4a_{n-1} + a_{n-1} + 7$
$a_{n+1} = 5a_{n-1} + 7 (*)$
Now I'm not sure what to next. Can I shift $(*)$ down to $a-n$.
Any help will be appreciated.
 [/INST] hint: add the two equations to get: $(a_n+b_n) = (a_{n-1} + b_{n-1}) + 1\to a_n+b_n = n+1$. Can you take it from here?
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