Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$ I'm considering the following problem from some course notes.


  Suppose the following is known about a discrete-time LTI system:
  
  
  Given the input $x[n]=(1/3)^n$ for all $n$, the system produces output $y[n]=(2/3)(1/3)^n$ for all $n$.
  Given the input $x[n]=(1/2)^nu[n]$, where $u[n]$ is the unit step, the output is $y[n]=\delta[n]+a(1/4)^nu[n]$.
  
  
  The problem asks for the value of $a$ and for the response $y[n]$
  given the input $x[n]=(1/6)^nu[n]$.


I thought from (1) I could determine the system function by taking a Fourier or Z transform, and then use that with (2) to deduce the value of $a$. For example, the Z transform of (2) says $Y(z)=H(z)X(z)$, where $Y$ and $X$ are easily computed (or obtainable from a table). If I could deduce $H$ from (1), I could get $a$.

But, as discussed here, the signals $(1/3)^n$ and $(2/3)(1/3)^n$ for all $n$ do not have Fourier or Z transforms, so I'm stuck. I could imagine there being a typo: perhaps $(1/3)^n$ should be $(1/3)^{|n|}$. But if that's not the case, I'm not sure how to proceed.
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. $(3)$.