Solve the non-homogenous recurrence $h_n = 3h_{n-1} + 3(2)^n, h_0 = 1$? I am trying to solve the non-homogenous recurrence $h_n = 3h_{n-1} + 3(2)^n, h_0 = 1$.
I can see that the solution to the homogenous $h_n = 3h_{n-1}$ is clearly $h_n = 3^n$. That being said, I have no idea how to solve $h_n - 3h_{n-1} = 3(2)^n$: I assume it's some solution of the sort $(c_1n + c_2)(2)^n$, but I don't know where we'd go from here! Can anyone give advice?

Welcome to MSE!
You could always use generating functions, but that machinery isn't necessary here. Instead, there's a standard trick. Write
$$t_n = \frac{h_n}{2^n}$$
Then after dividing both sides by $2^n$ your recurrence becomes
$$t_n = \frac{3}{2}t_{n-1} + 3$$
Do you see how to solve this recurrence for $t$? Do you see how that will solve your original problem too?

I hope this helps ^_^
