Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 9"

Revision as of 09:42, 30 September 2006

Problem

Revised statement

Let $a_{n}$ be a geometric sequence of complex numbers with $a_{0}=1024$ and $a_{10}=1$ , and let $S$ denote the infinite sum $S = a_{10}+a_{11}+a_{12}+...$ . If the sum of all possible distinct values of $S$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, compute the sum of the positive prime factors of $n$ .

Original statement

Let $a_{n}$ be a geometric sequence for $n\in\mathbb{Z}$ with $a_{0}=1024$ and $a_{10}=1$ . Let $S$ denote the infinite sum: $a_{10}+a_{11}+a_{12}+...$ . If the sum of all distinct values of $S$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then compute the sum of the positive prime factors of $n$ .

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Mock AIME 1 2006-2007

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 ===Original statement===
+Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</math> and <math>a_{10}=1</math>. Let <math>S</math> denote the infinite sum: <math>a_{10}+a_{11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then compute the sum of the positive prime factors of <math>n</math>.
 ==Solution==

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