<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>8.0.11 (Compound interest sequence)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>If {@N@} is invested at {@r@}% interest, compounded annually, then after \(n\) years the investment is worth \( a_n= {@N@}({@m@})^n \) pounds.</p><p><b>(a)</b> Find the first five terms of the sequence \( a_n\), each rounded to 2 d.p.</p><table style="margin-left:2em;" cellpadding="5">
<tbody>
<tr>
<td>\(a_1=\) [[input:ans1]]</td>
<td>[[validation:ans1]] </td>
<td>[[feedback:prt1]]</td>
</tr>
<tr>
<td>\(a_2=\) [[input:ans2]]</td>
<td>[[validation:ans2]]</td>
<td>[[feedback:prt2]]</td>
</tr>
<tr>
<td>\(a_3=\) [[input:ans3]]</td>
<td>[[validation:ans3]] </td>
<td>[[feedback:prt3]]</td>
</tr>
<tr>
<td>\(a_4=\) [[input:ans4]]</td>
<td> [[validation:ans4]]</td>
<td>[[feedback:prt4]]</td>
</tr>
<tr>
<td>\(a_5=\) [[input:ans5]]</td>
<td>[[validation:ans5]]</td>
<td>[[feedback:prt5]]</td>
</tr>
</tbody>
</table>
<br><p><b>(b)</b> Is the sequence convergent or divergent?</p><p>[[input:ans6]][[feedback:prt6]][[validation:ans6]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p><b>(a)</b>&nbsp;\( a_n= {@N@}({@m@})^n \)</p><p>For \( n= 1\), \( a_1= {@ta1@}.\)</p><p>For \( n= 2\), \( a_1= {@ta2@}.\)<br></p><p>For \( n= 3\), \( a_1= {@ta3@}.\)<br></p><p>For \( n= 4\), \( a_1= {@ta4@}.\)<br></p><p>For \( n= 5\), \( a_1= {@ta5@}.\)<br></p><p><b>(b)</b>&nbsp;Since \(({@m@})^{@n@}\) is of the form \(a^n\) with \(a&gt;0\), the terms will be increasing and unbounded, so \( \displaystyle \lim_{n\rightarrow \infty} a_n = {@lim@} \), i.e. the sequence is divergent.</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2022071300</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[N:rand([1,2,3,4,5])*1000;
r: rand([3,4,5,6,7]);
m: 1+float(r/100);

an: N*m^n
k:5;
S1:ev(makelist(k,k,1,k),simp);
S2:maplist(lambda([ex],ev(an,n=ex)),S1);
S3:ev(S2,simp);

lim: ev(limit(an,n,inf),simp);

ta1: S3[1]
ta2: decimalplaces(S3[2],2)
ta3: decimalplaces(S3[3],2)
ta4: decimalplaces(S3[4],2)
ta5: decimalplaces(S3[5],2)
ta:[["Convergent",false], ["Divergent",true]];
ta6:"Divergent"]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\( a_n= {@N@}({@m@})^n\) {@ta1@}, {@ta2@}, {@ta3@}, {@ta4@}, {@ta5@}</text>
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      <text/>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-2-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt6-1-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1+0.01</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2+0.01</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3+0.01</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4+0.01</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5+0.01</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value/>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt5-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
