<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>8.2.18 (Summing geometric progressions)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h3>Summing geometric progressions</h3>

<p>We will use the alternative name <b>geometric progression</b> to refer to a finite number of terms from a geometric sequence.</p>
<p>Adding up the first \(n\) terms of the geometric sequence given by \(u_n=ar^{n-1}\), we have
\[ 
\begin{align*} S_n &amp;= u_1+u_2+\cdots+u_n \\ &amp;= a + ar + ar^2 + \cdots + ar^{n-1}.
\end{align*} 
\]
Writing out \(S_n\) and \(r\times S_n\) in full,
\[
\begin{align*} S_n &amp;= a + ar + ar^2 + \cdots + ar^{n-1} \\ r\times S_n &amp;= \phantom{a +{}} ar + ar^2 + \cdots + ar^{n-1} + ar^n, \end{align*} 
\]
so we see that 
\[
\begin{align*} S_n - r S_n &amp;= a - ar^n \\ (1-r)S_n &amp;= a(1-r^n). \end{align*}
\]
This can be rearranged to get the formula for the sum of the first \(n\) terms:</p>
<br/>
<p>The sum of the first \(n\) terms of the geometric sequence given by \(u_n=ar^{n-1}\) is \[ S_n = \frac{a(1-r^n)}{1-r}. \]</p>
<br/>
<p>Note that the formula does not work if \(r=1\) since that would result in dividing by \(0\). But when \(r=1\) we just have \(S_n = a+a+a+\cdots+a=na\).</p>
<br/>
<h3>Example</h3>

<p>What is the sum of the geometric progression \(81+27+9+\cdots+\frac{1}{81}\)?</p>

<h3>Solution</h3>

<p>In case case, we see that the first term of the progression is \(a = 81\) and common ratio is \(r = \frac{27}{81} = \frac{1}{3}\). The final term is 
\[
\frac{1}{81} = ar^{n-1} = 81(\frac{1}{3})= 81(\frac{1}{3})^{n-1}.
\]
Therefore, 
\[
(\frac{1}{3})^{n-1} = \frac{1}{81^2}=\frac{1}{3^8}.
\]
Hence, \(n-1 = 8\) and so \(n=9\).
We are now in a position to use the formula above as follows
\[
S_9 = \frac{a(1-r^n)}{1-r} = \frac{81(1-\frac{1}{3^9})}{1-\frac{1}{3}} = \frac{9841}{81}
\]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
