The question I have been given is
Given that $z=2e^{i\frac{\pi}{7}}$, find the smallest positive integer of $k$ such that $z\times{z^2}\times{z^3}\times{\cdots}\times{z^k}$ is real, and state the value of $|z\times{z^2}\times{z^3}\times{\cdots}\times{z^k}|$ in this case.
A previous part of the question had me show that for any complex number $z=re^{i\theta}$, $z\times{z^2}\times{z^3}\times{\cdots}\times{z^k}=(re^{i\theta})^{\frac{k(k+1)}{2}}$, which I achieved using de Moivre's Theorem
Here's what I've tried doing;
Using Euler's Identity, I expanded the product to
$\cos{\left(\frac{k(k+1)\frac{\pi}{7}}{2}\right)}+i\sin{\left(\frac{k(k+1)\frac{\pi}{7}}{2}\right)}$
For a number to be real, I know that it's imaginary component must equal zero, so I figured that I would have to find the smallest positive integer of $k$ that satisfies the equation
$$\sin{\left(\frac{k(k+1)\frac{\pi}{7}}{2}\right)}=0$$
From here, I tried the following;
$$\therefore \frac{k(k+1)\frac{\pi}{7}}{2}=\arcsin{0}=0+a\pi\qquad a\in\mathbb{Z} $$
$$\therefore k(k+1)=\frac{2a\pi}{\frac{\pi}{7}}$$
$$\therefore k^2+k=14a$$
$$\therefore k^2+k-14a=0$$
$$\therefore k=\frac{-1\pm\sqrt{1-56a}}{2}$$
I'm not sure where to go from here. If $a$ was a defined constant, I could easily find $k$, however, because it isn't and I'm trying to find the smallest positive integer. According to the textbook, the solution is $k=6$, giving a modulus of $2^{21}$, but I do not understand how they came to this answer. I think I've gone about this the wrong way and there is probably a simpler method.
Edit: Fixed my dumb $\arcsin{0}$ mistake..
Best Answer
You are correct.
We have that $Arg((z)(z^2)(z^3)(z^4)...)=Arg(z)+2Arg(z)+3Arg(z)...\frac{1}{2}n(n+1)Arg(z)$
Triangular numbers are the simplest method.
We note that since $21$ is the first triangular number that is divisible by $7$, that is the value for which $Arg(z) | π$, and is thus real.
Because we have $z^{21}$ and $|z|=2$, it follows that $|z^{21}|=2^{21}$.
Good job!