# Mock AIME 1 2006-2007 Problems/Problem 9

## Contents

## Problem

### Revised statement

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 .

### Original statement

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 .

## Solutions

### Solution 1

Let the ratio of consecutive terms of the sequence be . Then we have by the given that so and , where can be any of the tenth roots of unity.

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

where .

Let ,

and

We seek , or the negative of the coefficient of divided by the coefficient of , which is and .

Therefore the answer is .