7040
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1
2011-02-01T00:00:01
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<div>
<br />
According to my calculator, I see that<br />
<br />
$$<br />
\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {15}+ \sqrt{16}} = 3.0000000000<br />
$$<br />
How could I prove that the answer is, indeed, exactly 3?<br />
<br />
<br />
</div>
<div class="framework">
<span style="font-style: italic;">Did you know ... ?</span>
<br />
<br />
Mathematicians often use calculating aids to help form an opinion concerning the likely numerical answer to a problem involving irrational numbers, but will always seek a full proof which does not rely on the calculator to complete the problem.</div>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Let<br></br>
$$X= \frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} +
\text{and so on up to}+\frac{1}{ \sqrt {15}+ \sqrt{16}} $$<br></br>
<br></br>
When surds appear in the denominator the first step is almost
always to rationalise the denominator. We see that<br></br>
$$<br></br>
X= \frac{1}{\sqrt{1}+ \sqrt{2}}\left(\frac{\sqrt{1}-
\sqrt{2}}{\sqrt{1}- \sqrt{2}}\right)+\frac{1}{\sqrt{2}+
\sqrt{3}}\left(\frac{\sqrt{2} -\sqrt{3}}{\sqrt{2}- \sqrt{3}}\right)
+ \dots+\frac{1}{ \sqrt {15}+ \sqrt{16}}\left( \frac{\sqrt
{15}-\sqrt{16}}{ \sqrt {15}-\sqrt{16}}\right)$$<br></br>
<br></br>
For each term the denominators multiply to $-1$. For example<br></br>
$$<br></br>
(\sqrt{2}+ \sqrt{3})(\sqrt{2}- \sqrt{3}) =
(\sqrt{2})^2-(\sqrt{3})^2= 2-3= -1<br></br>
$$<br></br>
Thus we have <br></br>
$$<br></br>
X= -(\sqrt{1}- \sqrt{2})-(\sqrt{2}+ \sqrt{3}) -\dots-(\sqrt
{15}-\sqrt{16})= -\sqrt{1}+\sqrt{16} = 3<br></br>
$$<br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Try rationalising the denominators.<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Let<br></br>
$$X= \frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} +
\text{and so on up to}+\frac{1}{ \sqrt {15}+ \sqrt{16}} $$<br></br>
<br></br>
When surds appear in the denominator the first step is almost
always to rationalise the denominator. We see that<br></br>
$$<br></br>
X= \frac{1}{\sqrt{1}+ \sqrt{2}}\left(\frac{\sqrt{1}-
\sqrt{2}}{\sqrt{1}- \sqrt{2}}\right)+\frac{1}{\sqrt{2}+
\sqrt{3}}\left(\frac{\sqrt{2} -\sqrt{3}}{\sqrt{2}- \sqrt{3}}\right)
+ \dots+\frac{1}{ \sqrt {15}+ \sqrt{16}}\left( \frac{\sqrt
{15}-\sqrt{16}}{ \sqrt {15}-\sqrt{16}}\right)$$<br></br>
<br></br>
For each term the denominators multiply to $-1$. For example<br></br>
$$<br></br>
(\sqrt{2}+ \sqrt{3})(\sqrt{2}- \sqrt{3}) =
(\sqrt{2})^2-(\sqrt{3})^2= 2-3= -1<br></br>
$$<br></br>
Thus we have <br></br>
$$<br></br>
X= -(\sqrt{1}- \sqrt{2})-(\sqrt{2}+ \sqrt{3}) -\dots-(\sqrt
{15}-\sqrt{16})= -\sqrt{1}+\sqrt{16} = 3<br></br>
$$<br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
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Weekly challenge 9: Quick Sum
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