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Suppose 1, a1, a2,..., a2008 are the 2009th roots of unity. Evaluate the sum from r = 1 to 2008 of r*(a_r + a_(2009 - r)).

1. 2009
2. 2008
3. 0
4. -2009

Correct answer: -2009

Solution

Reindexing pairs the linear coefficients so they cancel, leaving 2009 times the sum of the non-trivial roots, which is -1; hence -2009.

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