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1) Show the nth root of unity are the vertice of regular polygon

2) Find the formula for the perimeter of that polygon called "ln" and prove
$lim_nrightarrow infty l_n=2pi$

My attempt

Let $z=1$such that $zin mathbbC$. We need the nth root of the unity.

Let $win mathbbC$. such that $w^frac1n=z$

Then

$w_k=cos(fractheta+2kpin)+isin(fractheta+2kpin),,,,(1)$ for $k=0,1,...,n-1$

As $z=1$ then $theta=0$. Replacing in $(1)$ we have:

$w_k=cos(frac2kpin)+isin(frac2kpin)=e^ifrac2kpin,,,,(2)$

Here, i'm stuck.

I make a graphich representation for $k=4$ and is a polygin of four vertices. But for $n$ i'm stuck.

For the 2) question i don't have idea. Can someone help me?

You want to show that the angle between $e^2pi k i /n$ and $e^2pi (k+1)i/n$ is constant.

Note that if you divide these two complex numbers you get the resulting angle of rotation between the two. $$ frac e^2pi (k+1)i/ne^2pi ki /n= e^2pi i/n$$ which is the same for all $k$, that is they are vertices of a regular polygon, considering that they all have unit length.

For the side-length of the polygon you need to find the norm of the difference of two consecutive roots, for example $$|1-e^2pi i/n|$$

Multiply the result by n and let n goes to $infty$ to get your $2pi$

In De Moivre's formula, you see that the argument of two consecutive roots differ of $2pi/n$. Since the module of every root is equal to $1$, the points are distributed lie on the circumference $|z|=1$ and the angular distance between two consecutive points is $2pi/n$. Therefore the points are vertex of a regular $n$-agon. By the chord theorem, the length of the $n$-agon is $2sin(pi/n)$. Therefore $$ l_n=2nsin(pi/n)=2picdotfracsin(pi/n)pi/noversetnto inftyto 2pi. $$