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Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 6"

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<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
S &= \sum_{n=1}^{9800}a-b\sqrt{c} \\
 
S &= \sum_{n=1}^{9800}a-b\sqrt{c} \\
−
&= \sqrt{\frac{n+1}{2}}-\sqrt{\frac{n-1}{2}} \\
+
&= \sum_{n=1}^{9800}\sqrt{\frac{n+1}{2}}-\sqrt{\frac{n-1}{2}} \\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  

Revision as of 15:59, 28 November 2019

Problem

Let $S$ denote the value of the sum

\[\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}\]

$S$ can be expressed as $p + q \sqrt{r}$, where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Determine $p + q + r$.

Solution

Notice that $\sqrt{n + \sqrt{n^2 - 1}} = \frac{1}{\sqrt{2}}\sqrt{2n + 2\sqrt{(n+1)(n-1)}} = \frac{1}{\sqrt{2}}\left(\sqrt{n+1}+\sqrt{n-1}\right)$. Thus, we have

\[\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}\] \[= \sqrt{2}\sum_{n = 1}^{9800} \frac{1}{\sqrt{n+1}+\sqrt{n-1}}\] \[= \frac{1}{\sqrt{2}}\sum_{n = 1}^{9800} \left(\sqrt{n+1}-\sqrt{n-1}\right)\]

This is a telescoping series; note that when we expand the summation, all of the intermediary terms cancel, leaving us with $\frac{1}{\sqrt{2}}\left(\sqrt{9801}+\sqrt{9800}-\sqrt{1}-\sqrt{0}\right) = 70 + 49\sqrt{2}$, and $p+q+r=\boxed{121}$.


Solution 2

Simplifying the expression yields \begin{align*} S &= \sum_{n=1}^{9800}\frac{1}{\sqrt{n+\sqrt{n^2-1}}} \\ &= \sum_{n=1}^{9800}\frac{\sqrt{n+\sqrt{n^2-1}}}{n+\sqrt{n^2-1}} \\ &= \sum_{n=1}^{9800}(\frac{\sqrt{n+\sqrt{n^2-1}}}{n+\sqrt{n^2-1}})\cdot(\frac{n-\sqrt{n^2-1}}{n-\sqrt{n^2-1}}) \\ &= \sum_{n=1}^{9800}(\sqrt{n+\sqrt{n^2-1}})\cdot(\sqrt{n+\sqrt{n^2-1}})^2 \\ &= \sum_{n=1}^{9800}\sqrt{n-\sqrt{n^2-1}} \\ \end{align*} Now we can assume that \[\sqrt{n+\sqrt{n^2-1}}=a+b\sqrt{c}\] \[\sqrt{n-\sqrt{n^2-1}}=a-b\sqrt{c}\] for some $a$, $b$, $c$.

Squaring the first equation yields \[n+\sqrt{n^2-1}=a^2+b^2c+2ab\sqrt{c}\] which gives the system of equations \[n=a^2+b^2c\] \[\sqrt{n^2-1}=2ab\sqrt{c}\] calling them equations $A$ and $B$, respectively.

Also we have \[\frac{1}{\sqrt{n+\sqrt{n^2-1}}}=\sqrt{n-\sqrt{n^2-1}}\] \[\frac{1}{a+b\sqrt{c}}=a-b\sqrt{c}\] \[a^2-b^2c=1\] which obtains equation $C$.

Adding equations $A$ and $C$ yields \[2a^2=n+1\] \[a=\sqrt{\frac{n+1}{2}}\] Squaring equation $B$ and substituting yields \[4a^2b^2c=n^2-1\] \[2\cdot(n+1)\cdot b^2c=n^2-1\] \[b^2c=\frac{(n-1)(n+1)}{2\cdot(n+1)}\] \[b\sqrt{c}=\sqrt{\frac{n-1}{2}}\]

Thus we obtain the telescoping series \begin{align*} S &= \sum_{n=1}^{9800}a-b\sqrt{c} \\ &= \sum_{n=1}^{9800}\sqrt{\frac{n+1}{2}}-\sqrt{\frac{n-1}{2}} \\ \end{align*}

See Also

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by
Problem 5 		Followed by
Problem 7
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