1. Find the inverse z-transform of the following function:

X(z) = 3z / (z - 0.5)

Solution:
To find the inverse z-transform, we need to express the given function in terms of partial fraction expansion.

X(z) = 3z / (z - 0.5)
= 3z / (z - 0.5) * (z / z)
= 3z^2 / (z^2 - 0.5z)

Now, we can write the partial fraction expansion as:

X(z) = A / (z - 0.5) + B / z

To find the values of A and B, we need to multiply through by the denominators and equate coefficients:

3z^2 = A * z + B * (z - 0.5)

Comparing coefficients on both sides, we get:

A + B = 0 (coefficient of z^2 terms)
-0.5B = 3 (coefficient of z terms)

Solving these equations, we find A = -3 and B = 6.

Now, we can express X(z) as:

X(z) = -3 / (z - 0.5) + 6 / z

Taking the inverse z-transform of each term separately, we get:

x[n] = -3 * (0.5)^n + 6 * 1^n

Therefore, the inverse z-transform of X(z) is:

x[n] = -3 * (0.5)^n + 6

2. Find the inverse z-transform of the following function:

X(z) = z^2 / (z - 0.25)

Solution:
Similar to the previous problem, we need to express the given function in terms of partial fraction expansion.

X(z) = z^2 / (z - 0.25)
= z^2 / (z - 0.25) * (z / z)
= z^3 / (z^2 - 0.25z)

Now, we can write the partial fraction expansion as:

X(z) = A / (z - 0.25) + B / z

To find the values of A and B, we multiply through by the denominators and equate coefficients:

z^3 = A * z + B * (z - 0.25)

Comparing coefficients on both sides, we get:

A + B = 0 (coefficient of z^3 terms)
-0.25B = 1 (coefficient of z terms)

Solving these equations, we find A = -4 and B = 1.

Now, we can express X(z) as:

X(z) = -4 / (z - 0.25) + 1 / z

Taking the inverse z-transform of each term separately, we get:

x[n] = -4 * (0.25)^n + 1 * 1^n

Therefore, the inverse z-transform of X(z) is:

x[n] = -4 * (0.25)^n + 1