Find Z- inverse transform: 𝑍−1 [ 14𝑧(𝑧−4)(𝑧−7)].

To find the Z-inverse transform of 14𝑧(𝑧−4)(𝑧−7), we can use the linearity property of the Z-transform.

First, let's expand the expression: 14𝑧(𝑧−4)(𝑧−7) = 14𝑧(𝑧^2 - 11𝑧 + 28)

Next, we can use the Z-transform table to find the inverse transform of each term separately.

The Z-inverse transform of 𝑧 is 𝑛𝑢(𝑛), where 𝑢(𝑛) is the unit step function.

The Z-inverse transform of 𝑧^2 is 𝑛(n-1)u(n-2)/2!.

The Z-inverse transform of -11𝑧 is -11𝑛𝑢(𝑛).

The Z-inverse transform of 28 is 28𝑢(𝑛).

Now, we can combine these inverse transforms using the linearity property.

So, the Z-inverse transform of 14𝑧(𝑧−4)(𝑧−7) is: 14𝑛(n-1)u(n-2)/2! - 11𝑛𝑢(𝑛) + 28𝑢(𝑛)

This is the final answer for the Z-inverse transform of 14𝑧(𝑧−4)(𝑧−7).