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## Homework Statement

Show that if c is any nth root of unity other than unity itself that:

1 + c + c^2 + ... + c^(n-1) = 0

## Homework Equations

1 + z + z^2 + ... + z^n = (1 - z^(n+1)) / (1 - z)

## The Attempt at a Solution

c is an nth root of unity other than unity itself => (1-c) =/= 0.

so,

1 + c + c^2 + ... + c^(n-1) = (1 - c^n) / (1 - c) (= 0 by assumption)

hence,

(1 - c)(1 + c + c^2 + ... + c^(n-1)) = 0

so either (1 - c) = 0 or (1 + c + c^2 + ... + c^(n-1)) = 0

but (1 - c) =/= 0 by definition

so (1 + c + c^2 + ... + c^(n-1)) = 0