Let $A$ be an $n\times n$ matrix, $B$ be an $n\times m$ matrix, $C$ an $m \times m$ matrix, and consider the sum $$\sum_{k = 0}^{n-1} A^k B C^k$$ Is there any smart way to rewrite this sum in a way similar to the partial sums of the geometric series, namely for $a,b \in \mathbb{R}$ $$\sum_{k = 0}^{n-1} b a^k = b \frac{1-a^n}{1-a}$$

Let $A$ be an $n\times n$ matrix, $B$ be an $n\times m$ matrix, $C$ an $m \times m$ matrix, and consider the sum $$\sum_{k = 0}^{N-1} A^k B C^k.$$ Is there any smart way to rewrite this sum in a way similar to the partial sums of the geometric series; namely for $a,b \in \mathbb{R},$ $$\sum_{k = 0}^{N-1} b a^k = b \frac{1-a^N}{1-a}?$$