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    <name>
      <text>8.2.5 (Sigma notation examples)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Example</h3>
<ol type = "a">
<li>Write \(3+4+5+6+7+8+9\) using sigma notation.</li>
<li> Write \(4+9+16+\cdots+100\) using sigma notation.</li>
</ol>

<b>Solution</b>

<ol type = "a">
<li>There are 7 terms so we could use \(\displaystyle\sum_{n=1}^7\). We want the first term to be 3, so we should add up \(n+2\) each time, i.e.
\[ 
\sum_{n=1}^7 (n+2) = 3+4+5+6+7+8+9. 
\]</li>
<li>Note that these are all squares, so of the form \(n^2\). We start with \(n=2\) and end up with \(n=10\), so we have
\[ 
\sum_{n=2}^{10} n^2 = 4+9+16+\cdots+100. 
\]
What if we wanted to start with \(n=1\) instead? We would also need to change the end number and the formula:
\[ 
\sum_{n=1}^9 (n+1)^2 = 4+9+16+\cdots+100. 
\]
As this example shows, there is no unique way to write a sum using sigma notation.</li>
</ol>]]></text>
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