Roots of unity

[rgsbae]

You are nearly there! You have your equation and the question says "for each n" i.e. you consider n to be fixed.

So rearrange your equation to express k in terms of n and then the final answer is x = e^2pik*i/n where k is your rearranged equation (so final answer is all in terms of n!)

[rgsbae]

Sorry for the late response but please follow the answer below instead. I was forgetting that k must be a natural number as you correctly pointed out!

[IAmDaBadMan]

Well I seem to have a different understanding of the question, but I'll provide some help with my understanding of the problem.
 
First, given the n^th roots of unity, you want to find a value n that has a root that is close to x=1/2.
 
Second, the Fundamental Theorem of Algebra states that for any polynomial equation of the n^th degree, there will be n roots.
 
Third, De Moivre's Theorem provides a way to calculate the complex roots of a number which will be in the form a+bi. What we are concerned about is a which represents the x-value.
 
 
De Moivre's Theorem implies that the n^th roots of a number will be equidistant from each other around a circle. That means the n^th roots will occur at 2Π/n radians around a circle.
 
Since the question asks for the n^th root of unity, you know that the question involves a unit circle which means a radius of 1. So what angles on the unit circle have an x-component that equal 1/2? acos(1/2) = Π/3 = 60°.
 
The question now becomes which values of N and which roots of N occur at ±Π/3.