Originally Posted by
HallsofIvy If $\displaystyle a_n$ is the general solution to the associated homogeneous equation, $\displaystyle 4a_{n+2}- 4a_{n+1}+ a_n= 0$, and $\displaystyle b_n$ is any solution to the entire equation, $\displaystyle 4a_{n+2}- 4a_{n+1}+ a_n= (\frac{1}{3})^n$, then $\displaystyle a_n+ b_n$ is the general solution to the entire equation.
Do you know how to find the general solution to the associated homogeneous equation?
To find a solution to the entire equation, try [tex]a_n= A\left(\frac{1}{3}\right)^n[/itex].