Question

i$)$ Derive the $z-$transforms of the time function $t^2{e}^{-at}u(t).$ Do not use $z-$transform tables. Use the fact the $\frac{1}{1-z^{-1}}=1+z^{-1}+z^{-2}+z^{-3}$dots

ii$)$ Given $F(z)=\frac{z^3+2z+1}{(z-0\mathrm{.}1)(z^2+z+0\mathrm{.}5)},$ find $f(kT)$ using partial fraction expansion and tables.

iii$)$ Solve the linear difference equation $x(k+2)-\frac{3}{2}(k+1)+\frac{1}{2}x(k)=1(k)$ with the initial conditions $x(0)=1,x(1)=\frac{5}{2}$