Mock AIME 3 Pre 2005 Problems/Problem 6
Problem
Let 
denote the value of the sum
![\[\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}\]](http://latex.artofproblemsolving.com/0/4/2/04232c9e753b7168ec35810125ebb17786ddf7de.png)
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
$$ (Error compiling LaTeX. Unknown error_msg)\begin{align*}
\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}} &= \sqrt{2}\sum_{n = 1}^{9800} \frac{1}{\sqrt{n+1}+\sqrt{n-1}} \\
&= \frac{1}{\sqrt{2}}\sum_{n = 1}^{9800} \left(\sqrt{n+1}-\sqrt{n-1}\right) \$![$This is a [[telescope|telescoping]] series; note that when we expand the summation, all of the intermediary terms cancel, leaving us with$](http://latex.artofproblemsolving.com/1/8/d/18de58130a230b02b39094db1e39240d2324a0db.png)
\frac{1}{\sqrt{2}}\left(\sqrt{9801}+\sqrt{9800}-\sqrt{1}-\sqrt{0}\right) = 70 + 49\sqrt{2}
p+q+r=\boxed{121}$.
See Also
| Mock AIME 3 Pre 2005 (Problems, Source) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
