Determine the value of $\sum_{k=0}^n i^k$

Question

this is the problem:

Determine the value of the sum $\sum_{k=0}^n i^k$ for $n \in \Bbb N$.

I started to sum term by term but don't get its value

Answer 1

$S = i^0 + i^1 + i^2 + ... + i^{n-1} + i^n$

$Si = i^1 + i^2 + i^3 + ... + i^n + i^{n+1}$

$Si - S = i^{n+1} - i^0$

$S(i-1) = i^{n+1} - i^0$

$S = \dfrac{i^{n+1} - 1}{i-1}$

Answer 2

Hint: I want to know what $1 + i + i^2 + i^3 + ..... + i^n$ is.

So here is a question. What is $(1-i)(1 + i + i^2 + i^3 + ..... + i^n)$?

so if $(1-i)(1 + i + i^2 + i^3 + ..... + i^n)= M$ and if we assume $i \ne 1$ then that would mean $1 + i + i^2 + i^3 + ..... + i^n = \frac M{1-i}$.

This is a very common result that you will see a lot in your career. It's always fun to try to see if you can figure it out entirely on your own, but most of us had someone else show it to us and we all went "Oh, yeah! That's neat!"