Problem Find the ztransform in closed form of the number seq

Problem Find the z-transform, in closed form, of the number sequence generated by sampling, every T seconds the function f(t) whose Laplace transform is given by F(s) = 2(1 - e^-5s)/s(s = 2), T = 1 s

Solution

F(s)=2(1-e^-5)/(s(s+2))

=(1-e^-5)(1/s-1/(s+2))

As we know: z=e^(sT)=e^s

Henc . e^-5=z^-5

Z transform of 1/s=z/(z-1)

Z transfrom of 1/(s+2)=z/(z-e^-2)

F(z)=(1-z^-5)[z/(z-1)-z/(z-e^-2)]

F(z)=(1-z^-5)z(1-e^-2)/[(z-1)(z-e^-2)]