The question fits in the title: what tool(s) is/are needed to deduce this identity?
$$\sum\_{k=1}^ne^{ik}=\frac{(e^{in}-1)e^i}{e^i-1}.$$
This is the last step in my solution to another problem; I've tried playing around with the Binomial theorem to no avail and don't see what to try next. A good hint will suffice!