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

[Albert1]

$ 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}) $
$S_n=a_1+a_2+a_3+-------+a_n$
$find:\,\, S_{2012}$

 

[anemone]

Rationalizing the denominator of the expression \(\displaystyle a_n\) 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 \(\displaystyle S_{2012}\), we get:

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

\(\displaystyle S_{2012}=\frac{1}{2}-\frac{1}{2}(\sqrt{2013}-\sqrt{2012})\)

 

[Albert1]

Rationalizing the denominator of the expression \(\displaystyle a_n\) 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 \(\displaystyle S_{2012}\), we get:

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

\(\displaystyle S_{2012}=\frac{1}{2}-(\sqrt{2013}-\sqrt{2012})-------(last \,\, step)\)

your last step is incorrect ,a typo happens

 

[anemone]

your last step is incorrect ,a typo happens

Yep, you're right Albert...I left off \(\displaystyle \frac{1}{2}\) 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.

 

[CellCoree]

To simplify fractions using surds in the summation formula, we first need to understand what surds are. Surds are numbers that cannot be simplified into exact fractions, such as square roots, cube roots, etc. In the given summation formula, we can see that each term contains surds in the denominator.

To simplify these fractions, we can use the concept of rationalizing the denominator. This means multiplying the numerator and denominator by the same number to eliminate the surd in the denominator. In this case, we can multiply each term by the conjugate of the denominator, which is obtained by changing the sign between the two surds.

For example, for the first term, we can multiply both the numerator and denominator by $\sqrt{n+1} - \sqrt{n-1}$. This will result in the denominator becoming a difference of squares, which can be simplified.

After simplifying each term in the summation formula, we can then add them together to find the value of $S_{2012}$. This can be done by using a calculator or by simplifying the fractions further to obtain a final value.

In conclusion, to simplify fractions using surds in the summation formula, we need to rationalize the denominator by multiplying it with the conjugate of the surd. This will help us simplify the fractions and find the value of the given summation.