Why Does This Mathematical Expression Equal Zero?

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The mathematical expression in question equals zero because the terms represent points in the complex plane that form the vertices of a regular polygon. Each point acts as a vector originating from the center, and the symmetry of the polygon ensures that these vectors sum to zero. This conclusion holds true unless a specific condition is met, where the variable c is a multiple of q/r, causing all vertices to converge at a single point. Understanding this geometric interpretation clarifies why the term evaluates to zero. The discussion effectively illustrates the relationship between complex numbers and geometric shapes in mathematics.
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URGENT: Why does this tern = 0??

Can anyone tell me why the term in the bracket = 0??

where i= root -1 and all vars are integers

I'm at a loss :-/
 

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Each exponential in the sum represents a point in the complex plane and if you look at the terms carefully you will see that the set of points comprise the vertices of a regular polygon. Think of each point as being a vector from the origin to the given point. Being the vertices of a regular polygon, the vectors must add to zero! That is, unless c is a multiple of q/r in which case all the vertices converge to a single point.
 
Hehe, ok I think that makes sense.. thanks
 

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