Mock AIME 3 Pre 2005 Problems/Problem 6
Contents
Problem
Let denote the value of the sum
can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine .
Solution
Notice that . Thus, we have
This is a telescoping series; note that when we expand the summation, all of the intermediary terms cancel, leaving us with , and .
Solution 2
Simplifying the expression yields Now we can assume that for some , , .
Squaring the first equation yields which gives the system of equations calling them equations and , respectively.
Also we have which obtains equation .
Adding equations and yields Squaring equation and substituting yields
Thus we obtain the telescoping series
Simplifying the sum we are left with
Thus .
~ Nafer
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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