Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$
This problem tests the understanding of two basic properties of (discrete-time) LTI systems. First, for an input signal of the form $x[n]=z_0^n$, $-\infty&lt;n&lt;\infty$, the output is given by

$$y[n]=z_0^nH(z_0)\tag{1}$$

where $H(z)$ is the system's transfer function, and where it is assumed that $z_0$ is inside the region of convergence (ROC) of $H(z)$. [Note that $(1)$ is not true for $z_0$ outside the ROC.]

The second property is simply the $\mathcal{Z}$-domain input-output relationship

$$Y(z)=X(z)H(z)\tag{2}$$

Applying $(1)$ to your example you can conclude that

$$H\left(\frac{1}{3}\right)=\frac23\tag{3}$$

and from $(2)$ you can get an expression for $H(z)$ in terms of the constant $a$. This constant can be determined from Eq. $(3)$.