inverse Z-transform: 2z^2(-1/(z-.5)+1/(z-1))
you have done things right upto the part where you find the inverse. While calculating the inverse, as the text reads, the ROC is |z|&lt;1/2, which means that the system is left sided(think-s domain,causality). Accordingly you need to take the inverse of the function. You have used u(-n). However, when taking the inverse for a left sided function, we use u(-n-1) instead of u(-n). So, this is the wrong step in your solution. Once you have X(n), you can easily figure out Y(n) using the property (which you have already applied).

Some common questions-->

Why u(-n-1) and not u(-n)?
 -Try plotting graphs of u(n), u(-n) and u(-n-1). You will realise that u(-n-1) gives a left sided function.

Why specifically u(-n-1)..why not u(-n-a) where a is some fraction?
-Since we are dealing in discrete domain, we cannot have values other than integers.