I have very simple question about characteristic roots of recursion problems.
Lets say we have characteristic equation: $r^2 - 5r + 6 = 0$. It has roots $r=2$, $r=3$.
When plugging this into $a_n = Ar^n + Br^n$, is there a priority whether $r=2$ can plug into $A2^n$ or $B2^n$?
I sometimes plug $r=3$ into $A2^n$, sometimes into $B2^n$, and depending on which I get a different answer.
I am very confused by this. Any explanation would be great.
I think the source of your confusion may be in the expression $a_n = Ar^n + Br^n$, which makes it seem that you should use the same root. In fact, you should be using a different root for the two exponentials. The basic concept is that since $r=2$ is a solution to the characteristic equation, $a_n = A2^n$ is one possible solution to the recurrence; since $r=3$ is also a solution to the equation, $a_n = B3^n$ is also a solution to the recurrence. But because the recurrence is linear, you can add the two solutions to get $a_n = A2^n + B3^n$.
If you instead think of the general solution as $a_n = Ar_0{}^n + Br_1{}^n$ then you may avoid the confusion.