MHB How Do I Simplify Fractions Using Surds in the Summation Formula?

AI Thread Summary
The discussion focuses on simplifying the expression for the sequence \( a_n \) using surds and rationalizing the denominator. The final form of \( a_n \) is derived as \( -\frac{1}{2}(\sqrt{n+1}+\sqrt{n-1}-2\sqrt{n}) \), which reveals it as a telescoping series. To compute \( S_{2012} \), the participants derive the expression \( S_{2012} = -\frac{1}{2}(\sqrt{2013}-\sqrt{1})+\frac{1}{2}(\sqrt{2012}-0) \). A correction is noted regarding a typo in the final step, where a \( \frac{1}{2} \) was omitted. The discussion emphasizes the importance of accuracy in mathematical expressions.
Albert1
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$ a_n=(\dfrac{1}{\sqrt n+\sqrt {n-1}})\times(\dfrac{1}{\sqrt {n+1}+\sqrt {n-1}})\times(\dfrac{1}{\sqrt {n+1}+\sqrt n}) $
$S_n=a_1+a_2+a_3+-------+a_n$
$find:\,\, S_{2012}$
 
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Rationalizing the denominator of the expression $$a_n$$ we get:

$ a_n=\left(\dfrac{1}{\sqrt n+\sqrt {n-1}}\cdot \dfrac {\sqrt n-\sqrt {n-1}}{\sqrt n-\sqrt {n-1}}\right)\times\left(\dfrac{1}{\sqrt {n+1}+\sqrt {n-1}}\cdot \dfrac {\sqrt {n+1}-\sqrt {n-1}}{\sqrt {n+1}-\sqrt {n-1}}\right)\times\left(\dfrac{1}{\sqrt {n+1}+\sqrt {n}}\cdot \dfrac {\sqrt {n+1}-\sqrt {n}}{\sqrt {n+1}-\sqrt {n}}\right) $

$ a_n=\left(\dfrac {\sqrt {n}-\sqrt {n-1}}{1}\right)\times \left(\dfrac {\sqrt {n}-\sqrt {n+1}}{-1}\right) \times \left(\dfrac {\sqrt {n+1}-\sqrt {n-1}}{2}\right)$$ a_n=-\dfrac{1}{2}\left(\sqrt{n+1}+\sqrt{n-1}-2\sqrt{n}\right)$

$ a_n=-\dfrac{1}{2}\left(\sqrt{n+1}-\sqrt{n}\right)+\dfrac{1}{2}\left(\sqrt{n}-\sqrt{n-1}\right)$

This is clearly a telescoping series and to compute $$S_{2012}$$, we get:

$$S_{2012}=-\frac{1}{2}(\sqrt {2013}-\sqrt {1})+\frac{1}{2}(\sqrt{2012}-0)$$

$$S_{2012}=\frac{1}{2}-\frac{1}{2}(\sqrt{2013}-\sqrt{2012})$$
 
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anemone said:
Rationalizing the denominator of the expression $$a_n$$ we get:$ a_n=-\dfrac{1}{2}\left(\sqrt{n+1}+\sqrt{n-1}-2\sqrt{n}\right)$

$ a_n=-\dfrac{1}{2}\left(\sqrt{n+1}-\sqrt{n}\right)+\dfrac{1}{2}\left(\sqrt{n}-\sqrt{n-1}\right)$

This is clearly a telescoping series and to compute $$S_{2012}$$, we get:

$$S_{2012}=-\frac{1}{2}(\sqrt {2013}-\sqrt {1})+\frac{1}{2}(\sqrt{2012}-0)$$

$$S_{2012}=\frac{1}{2}-(\sqrt{2013}-\sqrt{2012})-------(last \,\, step)$$
your last step is incorrect ,a typo happens
 
Albert said:
your last step is incorrect ,a typo happens

Yep, you're right Albert...I left off $$\frac{1}{2}$$ in front of the surds, I'm sorry and I will fix my first post so that I get the correct answer to this problem.
 
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