7036
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1
2011-02-01T00:00:01
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<a href="http://nrich.maths.org/1404&part=">Warm-up
problem</a>
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<a href="http://nrich.maths.org/318&part=">Try this next</a>
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<a href="https://nrich.maths.org/discus/messages/27/150408.html">Ask
NRICH</a>
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<a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29">Read
all about it</a>
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<a href="http://nrich.maths.org/7039&part=solution">Last week's
solution</a>
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<div>
<br />
Find the sum of $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+
\sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {99}+
\sqrt{100}}.$$<br />
<br />
Can you invent any similar sums which have integer answers?<br />
</div>
<div class="framework">
<span style="font-style: italic;">Did you
know ... ?</span>
<br />
<br />
Whilst this series can be summed using elementary methods
mathematicians devise various ways in which the sums of series can
be analysed. These are explored in greater detail in university
analysis courses.</div>
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<span class="editorial">This challenge previously feature on NRICH.
Correct solutions were recieved from Charlene from Brunei, Kiang
from Singapore, Andre from Bucharest and Jing of Madras College.
Well done to all of you. Charlene's solution is given below. Not as
hard as it at first looks! The moral is not to be put off by
appearances.</span><br></br>
<br></br>
<span style="font-weight: bold;">A great discussion took place
concerning this challenge on</span> <a href="https://nrich.maths.org/discus/messages/27/150408.html">Ask
NRICH</a>. <br></br>
<br></br>
<span style="font-weight: bold;">The solution makes use of
'rationalising the denominator' in which the denominator of each
term is converted to an integer by multiplication with a factor
chosen so as to use the expression</span><br></br>
$$<br></br>
(x-y)(x+y) = x^2-y^2<br></br>
$$<br></br>
<br></br>
In detail, the numerator and denominator of the terms can be
multiplied to give a more convenient value as follows:<br></br>
<br></br>
\begin{eqnarray}&&\frac{1 \times(\sqrt{1} -
\sqrt{2})}{(\sqrt{1} + \sqrt{2})(\sqrt{1} - \sqrt{2})} + \frac{1
\times (\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} -
\sqrt{3})} + \dots + \frac{1 \times (\sqrt{99} -
\sqrt{100})}{(\sqrt{99} + \sqrt{100})(\sqrt{99} - \sqrt{100})}\\
&=& \frac{(\sqrt{1} - \sqrt{2})}{-1} + \frac{(\sqrt{2} -
\sqrt{3})}{-1} + \dots + \frac{(\sqrt{99} - \sqrt{100})}{-1}\\
&=& (-\sqrt{1} + \sqrt{2}) + (-\sqrt{2} + \sqrt{3}) + \dots
+ (-\sqrt{99} + \sqrt{100}) \\ &=& -\sqrt{1} + \sqrt{100}\\
&=& -1 + 10 \\ &=& 9 \end{eqnarray}<br></br></mdoxml>
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HI<br></br>
<br></br>
<br></br>
This will be $$block$$<br></br>
<br></br>
however $this is inline$ so it h<br></br>
hi<br></br></mdoxml>
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