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Let Z_r (r = 1, 2,..., 50) be the roots of the equation sum_(r=0)⁵⁰ Z^r = 0. Evaluate sum_(r=1)⁵⁰ 1/(Z_r - 1).

1. -85
2. -25
3. 25
4. 75

Correct answer: -25

Solution

The roots are the 51st roots of unity other than 1; the standard result sum 1/(Z_r - 1) over the (n-1) nontrivial n-th roots of unity equals -(n-1)/2, giving -50/2 = -25.

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