Given is the following discrete system
$\begin{align*} &x(k + 1) = Ax(k) + Bu(k)\\ &x(0) = x_0\;. \end{align*}$
How do we prove that the explicit solution formula for $x(k)$ (analogously to the variation of constants formula in the continuous time case) is
$A^kx_0+\sum_{j=1}^kA^{j-k}Bu(j-1)\;?$
Thanks a lot!