I. Find the z transform and ROC of each of the ff sequence

1.

ROC: All z, 1/2 <z and 1/4 < z

the total ROC is z > 1/2

II. Use the Z transform to perform the convolution of the following sequence

since 's z transform is

is equivalent to

using wolfram alpha to solve partial fractions

then simplify

the region of convergence is 1/3 < z and 1/2 < z

then the total ROC is 1/2

the inverse z transform is : using the time shifting property z^-1X(z) = u(n-1)

part III. Find the causal signal x(n) fo the following z transforms

using wolfram alpha

Making it into a z transform

the region of convergence is

-1/2 < z, -1 < z, and the entire plane of z

the total region of convergence is the entire plane of z

the inverse z transform is