Powers of a Toeplitz matrix
Solution 1:
Your matrix can be written as an exponential of a nilpotent matrix: $$F=e^{TN},\qquad N=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right).$$ Note that $N^2=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$ and $N^3=0$. Now, $$F^k=e^{kT N}=\mathbf 1+kTN+\frac{k^2T^2}{2}N^2,$$ which gives the result.
P.S. If you are uncomfortable with matrix exponentials, then simply use induction.