[SOLVED] Recurrence Relation

k3N70n

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

Solve the recurrence relation
[tex]a_{n+2}+3a_{n+1}+2a_n=3[/tex]

2. Relevant equations

add the homogeneous solution to the particular solution

3. The attempt at a solution

Characteristic: [itex]s^2+3s+2=0 \Rightarrow x=-2,-1[/itex]

Homogeneous: [itex]a_n=A(-2)^n+Bn(-1)^n[/itex]

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

Particular: [itex]a_n=C3^n[/itex]

then [tex]C3^{n+2}+3C3^{n+1}+2C3^n=3^n[/tex]

letting n=0 yeilds[tex]a_n=\frac{3^n}{20}[/tex] for the particular solution

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

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