So I have a home assignment and one of the questions is one that I know the solution of, but can't quite properly prove it using math terms.
The question goes as follows :
Let $\pmb S$ = $\{$ $\pmb i$ $+$ $\pmb i^2$ $+$ $\pmb i^3$ $+$ $\dots$ $+$ $\pmb i^n$ $\}$ where $n$ $\in$ $\mathbb N$.
Prove that $\pmb S$ $^4$ $\in$ $\mathbb R$.
Now I know that :
$\pmb i^{4n}$ $\pmb =$ $\pmb 1$
$\pmb i^{4n+1}$ $\pmb =$ $\pmb i$
$\pmb i^{4n+2}$ $\pmb =$ $\pmb -$ $\pmb 1$
$\pmb i^{4n+3}$ $\pmb =$ $\pmb -$ $\pmb i$
And that no matter what $n$ $\in$ $\mathbb N$ is chosen, I can always divide $\pmb i^{n}$ by the closest $\pmb i^{4n}$ and then be left with worst case scenario $\bigl(i^{3})^4$, but as pointed above any $\pmb i^{4n}$ will be equal to $\pmb 1$ $\in$ $\mathbb R$. It proves what I want to say, but it's kind of a mess since I believe there's a way to start from the right side and get to the left or vice versa.
If anyone could give me any pointers it would be greatly appreciated.
Thanks!
If $S_n=i+i^2+\cdots+i^n$, then, since $1+i+i^2+i^3=0$,$$S_n=\begin{cases}0&\text{ if $n$ is of the form }4k\\i^n&\text{ if $n$ is of the form }4k+1\\i^n(1+i)&\text{ if $n$ is of the form }4k+2\\i^{n+1}&\text{ if $n$ is of the form $4k+3$}\end{cases}$$and each of the numbers $0$, $i^n$, $i^n(1+i)$, and $i^{n+1}$ is such that its fourth power is real.