In a recent project, I came up with the following equality which turned out to be extremely useful for counting conjugacy classes in certain division algebras (I won't go into the details here, it's not really relevant).
The equality is a s follows:
Let $q,m,n\in\mathbb N$ be fixed non-zero nartual numbers. Then \begin{equation}\frac{q-1}{q}\:\cdot \sum_{\begin{smallmatrix}e=0\\n\text{ does not divide }e\end{smallmatrix}}^{m-1}q^{m-e+\lceil\frac{e}{n}\rceil}=q^m-q^{\lceil\frac{m}{n}\rceil}, \end{equation} where $\lceil\frac{a}{b}\rceil:=\min\lbrace x\in\mathbb Z\mid x\ge\frac{a}{b}\rbrace$ is the upper integer value of $\frac{a}{b}$.
The proof of this equality is pretty straightforward and I won't bore you with it.
My question here is this- Can anybody offer some intuition to why such an equality should hold?
I realize that this question is a little vague.. A good answer for me would be, for example, a counting problem that can be solved by either hands of the equality.
In any case, I was hoping that someone might have some idea or maybe has seen this appear somewhere before. I'm trying to generalize a result I have, and thought that maybe understanding why this equality keeps popping up will be helpful in understanding the general problem.
Thank you.