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
\begin{align*} S &= \sum_{n=1}^{9800}\frac{1}{\sqrt{n+\sqrt{n^1-1}}} \\ &= \sum_{n=1}^{9800}\frac{\sqrt{n+\sqrt{n^1-1}}}{n+\sqrt{n^1-1}} \\ &= \sum_{n=1}^{9800}(\frac{\sqrt{n+\sqrt{n^1-1}}}{n+\sqrt{n^1-1}})\cdot(\frac{n-\sqrt{n^2-1}}{n-\sqrt{n^2-1}}) \\ &= \sum_{n=1}^{9800}\frac{(\sqrt{n+\sqrt{n^1-1})\cdot(\sqrt{n+\sqrt{n^1-1})^2}{\sqrt{n+\sqrt{n^1-1}}} \\ \end{align*} (Error compiling LaTeX. Unknown error_msg)
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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