Mock AIME 1 2006-2007 Problems/Problem 9
Let be a geometric sequence of complex numbers with and , and let denote the infinite sum . If the sum of all possible distinct values of is where and are relatively prime positive integers, compute the sum of the positive prime factors of .
Let be a geometric sequence for with and . Let denote the infinite sum: . If the sum of all distinct values of is where and are relatively prime positive integers, then compute the sum of the positive prime factors of .
Then the sum has value . Different choices of clearly lead to different values for , so we don't need to worry about the distinctness condition in the problem. Then the value we want is . Now, recall that if are the th roots of unity then for any integer , is 0 unless in which case it is 1. Thus this simplifies to
We seek , or the negative of the coefficient of divided by the coefficient of , which is and .
Therefore the answer is .