I need to determine the solutions for a recursion formula by using the amplification
$$ax_n - bx_{n+1} = cx_{n+2}$$
where $a,b,c$ are constants and $x_n,x_{n+1},x_{n+2}$ are iterations.
So far I have that the amplification factor is $A=\frac{x_{n+1}}{x_n}$. So substituting this in I get $A^2 - \frac{a}{c} = 0$. However I don't think this is right and don't want to continue down the wrong path so any help would be much appreciated.
You assume that the solution is proportional to $A^n$ where $A$ is the amplification factor. You also have $A=\frac {x_{n+2}}{x_{n+1}}$. Now substitute that in and you get $a+bA=cA^2$. Solve the quadratic in $A$, take the greater root, and you have it.