Find the inverse Z-transform of(𝒂) 𝒛𝟑−𝟐𝟎𝒛(𝒛−𝟐)𝟐(𝒛−𝟒)

The inverse Z-transform can be found using the method of partial fraction expansion.

The given function is F(z) = z^3 - 20z / (z-2)^2 * (z-4)

First, we need to express F(z) as a sum of simpler fractions.

Let's write F(z) as:

F(z) = A/(z-2) + B/(z-2)^2 + C/(z-4)

Multiplying through by the denominator to clear the fractions gives:

z^3 - 20z = A*(z-2)(z-4) + B(z-4) + C*(z-2)^2

We can solve for A, B, and C by substituting convenient values for z.

For example, setting z = 2 gives B = -12.

Setting z = 4 gives A = 4.

Substituting these values back into the equation and simplifying gives C = 2.

So, F(z) = 4/(z-2) - 12/(z-2)^2 + 2/(z-4)

Now, we can find the inverse Z-transform of each term separately.

The inverse Z-transform of 1/(z-a) is a^n, and the inverse Z-transform of n/(z-a)^(n+1) is a^n * u[n-1].

So, the inverse Z-transform of F(z) is:

f[n] = 42^n - 122^nu[n-1] + 24^n

This is the inverse Z-transform of the given function.