How can we show by mathematical induction that the following holds for $ n \ge 0$ and $a \ne 1$?
$ 1 + a + a^2 + a^3 + \ldots + a^n = \frac{a^{n+1}-1}{a-1}$
I understand the principle of mathematical induction, but I've no idea how to apply it here. I know in general I have to prove it for $n = 1$ and then assume $n = k$ is correct. Then prove $n = k+1$ is true. But what about $a$?
I've watched a load of YouTube videos on the subject, and I've read a few questions here but it's not helping. The videos make sense while I'm watching them, but I don't know how to apply it. This question appeared on my discrete math exam last week. I did not do well. I think I'm missing something fundamental in my understanding of this subject.