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I tried my best, but I am totally clueless about it. Worse thing is we were supposed to arrive at the answer in approximately $ 2 $ minutes. The correct answer is $ 8 $, right? Can you kindly explain how to arrive at it? I hope it won’t be too much bother. Thank you.

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  • $\begingroup$ Are you sure you've got the question right? If the terms were multiplied, it would be easy. $\endgroup$ Feb 22 '13 at 6:39
  • $\begingroup$ @JanDvorak Sorry for the delay in replying to you.Yes, I have got the question right.I double-check things before putting them up on this reputed maths forum. $\endgroup$
    – Ivy Mike
    Feb 22 '13 at 12:18
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Note that

$$ \frac{1}{\sqrt{n} + \sqrt{n+1}} = \sqrt{n+1} - \sqrt{n} $$

Then you have a telescoping sum

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  • $\begingroup$ Thanks Muzzlator for your fine answer. $\endgroup$
    – Ivy Mike
    Feb 22 '13 at 12:19
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As \begin{align} \frac{1}{\sqrt{i}+\sqrt{i+1}} = \frac{\sqrt{i}-\sqrt{i+1}}{\sqrt{i}-\sqrt{i+1}} \cdot \frac{1}{\sqrt{i}+\sqrt{i+1}} = \frac{\sqrt{i}-\sqrt{i+1}}{-1} = {\sqrt{i+1}-\sqrt{i}} \end{align} Thus \begin{align} \sum_{i=1}^{80} \frac{1}{\sqrt{i}+\sqrt{i+1}} = \sum_{i=1}^{80} {\sqrt{i+1}-\sqrt{i}} = \sqrt{80+1} -\sqrt{1} = 8 \end{align}

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  • $\begingroup$ A prof of mine always said, that "text is the shortest connection between two formulas" ;) $\endgroup$
    – Thomas
    Feb 22 '13 at 6:51
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$$\frac{\sqrt{2} - \sqrt{1} }{(2-1)} +\ldots +\frac{\sqrt{81} - \sqrt{80} }{(81-80)}=\sqrt{81}-\sqrt{1}=9-1=8 $$

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  • $\begingroup$ Thanks for your answer Jim. $\endgroup$
    – Ivy Mike
    Feb 22 '13 at 12:23

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