It just came to me that the most intuitive way (though not rigorous mathematically) is that those points on the complex plane form a regular n-polygon. Summing them up is equivalent to summing up those vectors on the complex plane, which should be zero by symmetry.

Below is an example for n = 5.

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You may also think of them as orthogonality relation (that’s why I call them orthogonality relation) for the representation of cyclic group Z_n, if you have background knowledge in group theory

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They are geometric series x + x^2 +x^3 +x^4 + .. +x^n with x= e^(2pi k/n)

Do you know how to sum geometric series? Should have learned this in high school.

ans