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<h1 align="center">Geometric Sequences and Sums</h1>
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<h2>Sequence</h2>
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<p>A Sequence is a set of things (usually numbers) that are in order. </p>
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<p class="center"><img src="images/sequence.svg" alt="Sequence" />
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</p>
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<h2>Geometric Sequences</h2>
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<p>In a <b>Geometric Sequence</b> each term is found by <b>multiplying</b> the previous term by a <b>constant</b>.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="simple">
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<table align="center">
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<tr>
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<td><font size="+1">1, 2, 4, 8, 16, 32, 64, 128, 256</font><font size="+1" class="large">, ...</font></td>
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</tr>
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</table>
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</div>
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<p align="center">This sequence has a factor of 2 between each number.</p>
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<p align="center">Each term (except the first term) is found by <b>multiplying</b> the previous term by <b>2</b>.</p>
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<p class="center"><img src="images/geometric-sequence-2.svg" alt="geometric sequence 1,2,4,8,16," /></p>
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</div>
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<p> </p>
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<p><b>In General</b> we write a Geometric Sequence like this:</p>
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<p align="center" class="large">{a, ar, ar<sup>2</sup>, ar<sup>3</sup>, ... }</p>
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<p>where:</p>
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<ul>
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<li><b>a</b> is the first term, and </li>
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<li><b>r</b> is the factor between the terms (called the <b>"common ratio"</b>)</li>
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</ul><p> </p>
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<div class="example">
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<h3>Example: {1,2,4,8,...}</h3>
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<p>The sequence starts at 1 and doubles each time, so</p>
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<ul>
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<li><b>a=1</b> (the first term) </li>
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<li><b>r=2</b> (the "common ratio" between terms is a doubling)</li>
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</ul>
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<p>And we get:</p>
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<p align="center" class="larger"><span class="larger">{a, ar, ar<sup>2</sup>, ar<sup>3</sup>, ... }</span></p>
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<p align="center" class="larger">= {1, 1×2, 1×2<sup>2</sup>, 1×2<sup>3</sup>, ... } </p>
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<p align="center" class="larger">= {1, 2, 4, 8, ... }</p>
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</div><p> </p>
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<p class="larger">But be careful, <b>r</b> should not be 0: </p>
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<ul>
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<li>When <b>r=0</b>, we get the sequence {a,0,0,...} which is not geometric</li>
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</ul>
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<h2>The Rule</h2>
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<p>We can also calculate <b>any term</b> using the Rule:</p>
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<div class="center80">
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<p align="center"><span class="large">x<sub>n</sub> = ar<sup>(n-1)</sup></span></p>
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<p align="center">(We use "n-1" because <span class="large">ar<sup>0</sup></span> is for the 1st term)</p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example:</h3>
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<div class="simple">
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<table align="center">
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<tr>
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<td><font size="+1"> 10, 30, 90, 270, 810, 2430, </font><font size="+1" class="large"> ...</font></td>
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</tr>
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</table>
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</div>
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<p align="center">This sequence has a factor of 3 between each number.</p>
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<p>The values of <b>a</b> and <b>r</b> are:</p>
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<ul>
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<li><b>a = 10</b> (the first term) </li>
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<li><b>r = 3</b> (the "common ratio")</li>
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</ul>
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<p>The Rule for any term is:</p>
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<p align="center" class="larger"><span class="large">x<sub>n</sub> = 10 × 3<sup>(n-1)</sup></span></p>
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<p>So, the <b>4th</b> term is:</p>
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<p align="center" class="larger">x<sub>4</sub> = 10<span class="large">×</span>3<sup>(4-1)</sup> = 10<span class="large">×</span>3<sup>3</sup> = 10<span class="large">×</span>27 = 270</p>
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<p>And the <b>10th</b> term is:</p>
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<p align="center" class="larger">x<sub>10 </sub>= 10<span class="large">×</span>3<sup>(10-1)</sup> = 10<span class="large">×</span>3<sup>9</sup> = 10<span class="large">×</span>19683 = 196830</p>
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</div>
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<p> </p>
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<p>A Geometric Sequence can also have <b>smaller and smaller</b> values:</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="simple">
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<table align="center">
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<tr>
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<td><font size="+1">4, 2, 1, 0.5, 0.25, .</font><font size="+1" class="large">..</font></td>
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</tr>
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</table>
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</div>
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<p align="center">This sequence has a factor of 0.5 (a half) between each number.<br />
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</p>
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<p align="center">Its Rule is <b>x<sub>n</sub> = 4 × (0.5)<sup>n-1</sup></b></p>
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</div>
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<h2>Why "Geometric" Sequence? </h2>
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<p>Because it is like increasing the dimensions in <b>geometry</b>:</p>
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<table border="0" align="center">
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<tr>
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<td rowspan="4"><img src="images/geometric-sequence.gif" alt="Geometric Sequence" width="72" height="217" /></td>
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<td height="55">a line is 1-dimensional and has a length of <span class="large">r</span></td>
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</tr>
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<tr>
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<td height="55">in 2 dimensions a square has an area of <span class="large">r<sup>2</sup></span></td>
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</tr>
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<tr>
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<td height="55">in 3 dimensions a cube has volume <span class="large">r<sup>3</sup></span></td>
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</tr>
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<tr>
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<td height="55">etc (yes we can have 4 and more dimensions in mathematics). </td>
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</tr>
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</table>
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<p> </p>
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<div class="words">
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<p>Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)</p>
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</div>
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<h2>Summing a Geometric Series</h2>
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<p><b>To sum these:</b></p>
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<p align="center"><span class="large">a + ar + ar<sup>2</sup> + ... + ar<sup>(n-1)</sup></span> </p>
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<p align="center">(Each term is <span class="large">ar<sup>k</sup></span>, where k starts at 0 and goes up to n-1)</p>
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<p><b>We can use this handy formula:</b></p>
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<p align="center"><img src="images/partial-sum-i.gif" alt="Sigma" width="195" height="54" /><br />
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<br />
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<b>a</b> is the first term <br />
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<b>r</b> is the <b>"common ratio"</b> between terms <br />
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<b>n</b> is the number of terms </p>
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<div class="center80">
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<p><i>What is that funny Σ symbol?</i> It is called <a href="sigma-notation.html">Sigma Notation</a></p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/sigma.gif" alt="Sigma" width="32" height="34" /></td>
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<td>(called Sigma) means "sum up"</td>
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</tr>
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</table>
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<p>And below and above it are shown the starting and ending values:</p>
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<p align="center"><img src="images/sigma-notation.svg" alt="Sigma Notation" /></p>
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<p align="center">It says "Sum up <i><b>n</b></i> where <i><b>n</b></i> goes from 1 to 4. Answer=<b>10</b></p>
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</div>
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<p>The formula is easy to use ... just "plug in" the values of <b>a</b>, <b>r</b> and <b>n</b></p>
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<div class="example">
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<h3>Example: Sum the first 4 terms of</h3>
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<div class="simple">
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<table align="center">
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<tr>
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<td><font size="+1"> 10, 30, 90, 270, 810, 2430, </font><font size="+1" class="large"> ...</font></td>
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</tr>
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</table>
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</div>
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<p align="center">This sequence has a factor of 3 between each number.</p>
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<p> </p>
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<p>The values of <b>a</b>, <b>r</b> and <b>n</b> are:</p>
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<ul>
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<li><b>a = 10</b> (the first term) </li>
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<li><b>r = 3</b> (the "common ratio")</li>
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<li><b>n = 4</b> (we want to sum the first 4 terms)</li>
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</ul>
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<p>So:</p>
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<p align="center"><img src="images/partial-sum-i.gif" alt="Sigma" width="195" height="54" /></p>
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<p>Becomes:</p>
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<p align="center"><img src="images/partial-sum-k2.gif" alt="Sigma" width="290" height="57" /></p>
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<p>You can check it yourself:</p>
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<p align="center" class="larger">10 + 30 + 90 + 270 = 400</p>
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<p>And, yes, it is easier to just add them <i>in this example</i>, as there are only 4 terms. But imagine adding 50 terms ... then the formula is much easier.</p>
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</div>
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<h2>Using the Formula</h2>
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<p>Let's see the formula in action:</p>
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<div class="example">
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<h3>Example: Grains of Rice on a Chess Board</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><a href="../games/chess.html"><img src="../images/chess-board.gif" alt="chess board" width="150" height="149" /></a></p>
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<p>On the page <a href="../binary-digits.html">Binary Digits</a> we give an example of grains of rice on a chess board. The question is asked: </p>
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<p>When we place rice on a chess board:</p>
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<ul>
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<li>1 grain on the first square, </li>
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<li>2 grains on the second square, </li>
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<li>4 grains on the third and so on, </li>
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<li>...</li>
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</ul>
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<p align="center">... <b>doubling</b> the grains of rice on each square ... </p>
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<p align="center"><b>... how many grains of rice in total?</b></p>
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<p>So we have:</p>
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<ul>
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<li><b>a = 1</b> (the first term)</li>
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<li><b>r = 2</b> (doubles each time)</li>
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<li><b>n = 64</b> (64 squares on a chess board)</li>
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</ul>
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<p>So:</p>
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<p align="center"><img src="images/partial-sum-i.gif" alt="Sigma" width="195" height="54" /></p>
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<p>Becomes:</p>
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<p align="center"><img src="images/partial-sum-j2.gif" alt="Sigma" width="204" height="58" /></p>
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<p> </p>
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<p align="center" class="large">= <span class="intbl"><em>1−2<sup>64</sup></em><strong>−1</strong></span> = 2<sup>64</sup> − 1 </p>
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<p align="center" class="large">= 18,446,744,073,709,551,615 </p>
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<p>Which was exactly the result we got on the <a href="../binary-digits.html">Binary Digits</a> page (thank goodness!)</p>
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</div>
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<p>And another example, this time with <b>r</b> less than 1:</p>
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<div class="example">
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<h3>Example: Add up the first 10 terms of the Geometric Sequence that halves each time:</h3>
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<h3 align="center"> { 1/2, 1/4, 1/8, 1/16, ... }</h3>
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<p>The values of <b>a</b>, <b>r</b> and <b>n</b> are:</p>
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<ul>
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<li><b>a = ½</b> (the first term)</li>
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<li><b>r = ½</b> (halves each time)</li>
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<li><b>n = 10</b> (10 terms to add)</li>
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</ul>
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<p>So:</p>
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<p align="center"><img src="images/partial-sum-i.gif" alt="Sigma" width="195" height="54" /></p>
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<p>Becomes:</p>
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<p align="center"><img src="images/partial-sum-i2.gif" alt="Sigma" width="239" height="180" /></p>
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<p align="center" class="larger">Very close to 1.</p>
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<p><i>(Question: if we continue to increase <span class="large">n</span>, what happens?)</i></p>
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</div>
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<h2>Why Does the Formula Work?</h2>
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<p>Let's see <b>why</b> the formula works, because we get to use an interesting "trick" which is worth knowing.</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>First</b>, call the whole sum <b>"S"</b>:</span><span class="right"><span class="large"> S = a + ar + ar<sup>2</sup> + ... + ar<sup>(n−2)</sup></span><span class="large">+ ar<sup>(n−1)</sup></span></span></div>
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<div class="row"><span class="left"><b>Next</b>, multiply <b>S</b> by <b>r</b>:</span><span class="right"><span class="large">S·r = ar + ar<sup>2</sup> + ar<sup>3</sup> + ... + ar<sup>(n−1)</sup> + ar<sup>n</sup></span></span></div>
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</div>
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<p align="center" class="larger">Notice that <span class="large">S</span> and <span class="large">S·r</span> are similar?</p>
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<p class="larger">Now <b>subtract</b> them!</p>
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<p align="center"><img src="images/geometric-sum-proof.svg" alt="Proof" /></p>
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<p align="center"><b>Wow! All the terms in the middle neatly cancel out. </b><br />
|
||
(Which is a neat trick)</p>
|
||
<p>By subtracting <span class="large">S·r</span> from <span class="large">S</span> we get a simple result:</p>
|
||
<div class="center80">
|
||
<p align="center" class="large">S − S·r = a − ar<sup>n</sup></p>
|
||
</div>
|
||
<p>Let's rearrange it to find <span class="large">S</span>:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Factor out <b>S</b> and <b>a</b>:</span><span class="right"><span class="larger">S(1<span class="large">−</span>r) = a(1<span class="large">−</span>r<sup>n</sup>)</span></span></div>
|
||
<div class="row"><span class="left">Divide by <b>(1−r)</b>:</span><span class="right"><span class="larger">S = <span class="intbl">
|
||
<em>a(1<span class="large">−</span>r<sup>n</sup>)</em>
|
||
<strong>(1<span class="large">−</span>r)</strong>
|
||
</span></span></span></div>
|
||
</div>
|
||
<p>Which is our formula (ta-da!):</p>
|
||
<p align="center"><img src="images/partial-sum-i.gif" alt="Sigma" width="195" height="54" /></p>
|
||
<p> </p>
|
||
<h2>Infinite Geometric Series</h2>
|
||
<p>So what happens when <span class="large">n</span> goes to <b>infinity</b>? </p>
|
||
<p>We can use this formula: </p>
|
||
<p align="center"><img src="images/geometric-infinite-sum.gif" alt="Sigma" width="189" height="55" /></p>
|
||
<p>But <b>be careful</b>:</p>
|
||
<div class="center80">
|
||
<p align="center" class="larger"><b>r</b> must be between (but not including) <b>−1 and 1</b></p>
|
||
<p align="center">and <b>r should not be 0</b> because the sequence {a,0,0,...} is not geometric</p>
|
||
</div>
|
||
<p>So our infnite geometric series has a <b>finite sum</b> when the ratio is less than 1 (and greater than −1)</p>
|
||
<p>Let's bring back our previous example, and see what happens:</p>
|
||
<div class="example">
|
||
<h3>Example: Add up ALL the terms of the Geometric Sequence that halves each time:</h3>
|
||
<h3 align="center"> { <span class="intbl"><em>1</em><strong>2</strong></span>, <span class="intbl"><em>1</em><strong>4</strong></span>, <span class="intbl"><em>1</em><strong>8</strong></span>, <span class="intbl"><em>1</em><strong>16</strong></span>, ... }</h3>
|
||
<p>We have:</p>
|
||
<ul>
|
||
<li><b>a = ½</b> (the first term)</li>
|
||
<li><b>r = ½</b> (halves each time)</li>
|
||
</ul>
|
||
<p>And so:</p>
|
||
<p align="center"><img src="images/geometric-infinite-sum-b.gif" alt="Sigma" width="225" height="62" /></p>
|
||
<p align="center" class="large">= <span class="intbl"><em>½×1</em><strong>½</strong></span> = 1</p>
|
||
<p>Yes, adding <b><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> + ...</b> etc equals <b>exactly 1</b>.</p>
|
||
</div>
|
||
<table border="0" align="center">
|
||
<tr>
|
||
<td align="center"><p>Don't believe me? Just look at this square:</p>
|
||
<p>By adding up <b><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>8</strong></span> + ...</b></p>
|
||
<p>we end up with the whole thing!</p></td>
|
||
<td align="center"> </td>
|
||
<td><span class="center"><img src="images/infinite-series-1-2n.svg" alt="Sum of 1/2^n as boxes" /></span></td>
|
||
</tr>
|
||
</table>
|
||
<h2>Recurring Decimal</h2>
|
||
<p>On another page we asked <a href="../9recurring.html">"Does 0.999... equal 1?"</a>, well, let us see if we can calculate it:</p>
|
||
<div class="example">
|
||
<h3>Example: Calculate 0.999...</h3>
|
||
<p>We can write a recurring decimal as a sum like this:</p>
|
||
<p align="center"><img src="images/geometric-infinite-sum-d.gif" alt="Sigma" width="398" height="120" /> </p>
|
||
<p>And now we can use the formula:</p>
|
||
<p align="center"><img src="images/geometric-infinite-sum-e.gif" alt="Sigma" width="403" height="54" /></p>
|
||
<p align="center"> </p>
|
||
<p>Yes! 0.999... <i><b>does</b></i> equal 1.</p>
|
||
</div>
|
||
<p> </p>
|
||
<p>So there we have it ... Geometric Sequences (and their sums) can do all sorts of amazing and powerful things.</p>
|
||
|
||
<div class="questions">
|
||
|
||
<script type="text/javascript">getQ(1755, 608, 609, 610, 611, 8300, 1256, 1257, 1258, 1255, 176);</script>
|
||
|
||
</div>
|
||
|
||
<div class="related"><a href="sequences-series.html">Sequences</a> <a href="sequences-sums-arithmetic.html">Arithmetic Sequences and Sums</a> <a href="sigma-notation.html">Sigma Notation</a> <a href="index.html">Algebra Index</a></div>
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