## [SOLVED] Recurrence Relation

1. The problem statement, all variables and given/known data

Solve the recurrence relation
$$a_{n+2}+3a_{n+1}+2a_n=3$$

2. Relevant equations

add the homogeneous solution to the particular solution

3. The attempt at a solution

Characteristic: $s^2+3s+2=0 \Rightarrow x=-2,-1$

Homogeneous: $a_n=A(-2)^n+Bn(-1)^n$

(*here is where I'm not sure if I've gone wrong. Should it be
$a_n=A(-1)^n+Bn(-2)^n$ instead? How would I know?*)
anyway, not sure I've right up to here but onwards->

Particular: $a_n=C3^n$

then $$C3^{n+2}+3C3^{n+1}+2C3^n=3^n$$

letting n=0 yeilds$$a_n=\frac{3^n}{20}$$ for the particular solution

then the solution is $$a_n=A(-2)^n+Bn(-1)^n+\frac{3^n}{20}$$

solving for A and B I got A=-1/20 and B=-3/4 and putting it all together seemed to give me nonsense.

Anyway, as you can probably see I'm not really understanding what's happening here. I would really appreciate nudge in the right direction.

Thank you kindly for you time
-kentt