Help Solving a Difference Equation
You seem to have figured out that the system is actually given by

$$y[n]=\frac14 y[n-1]+x[n],\quad \forall n\tag{1}$$

Since you're looking for the zero-state output, you can simply compute the system's impulse response $h[n]$ by applying $x[n]=\delta[n]$ as an input signal (with $y[n-1]=0$), and compute the response to any other input signal using this impulse response. For $x[n]=\delta[n+m]$, the response is simply $h[n+m]$.

The recursion for the impulse response is

$$\begin{align}h[0]&amp;=1\\
h[1]&amp;=\frac14\\
h[2]&amp;=\left(\frac14\right)^2\\
&amp;\vdots\end{align}\\$$

I'm sure you can take it from here and derive a general expression for $h[n]$.