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<h1 align="center">Sequences - Finding a Rule</h1>
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<p class="larger" align="center">To find a missing number in a Sequence, first we must have a<b> Rule</b></p>
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<h2>Sequence</h2>
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<p>A <a href="sequences-series.html">Sequence</a> 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"></p>
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<p>Each number in the sequence is called a <b>term</b> (or sometimes "element" or "member"), read <a href="sequences-series.html">Sequences and Series</a> for a more in-depth discussion.</p>
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<h2>Finding Missing Numbers</h2>
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<p>To find a missing number, first find a <b>Rule</b> behind the Sequence. </p>
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<p>Sometimes we can just look at the numbers and see a pattern:</p>
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<div class="example">
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<h3>Example: 1, 4, 9, 16, ?</h3>
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<p>Answer: they are <a href="../square-root.html">Squares</a> (1<sup>2</sup>=1, 2<sup>2</sup>=4, 3<sup>2</sup>=9, 4<sup>2</sup>=16, ...)</p>
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<p class="larger" align="center">Rule: <b>x<sub>n</sub> = n<sup>2</sup></b></p>
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<p>Sequence: 1, 4, 9, 16, <b>25, 36, 49, ...</b></p>
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</div>
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<p>Did you see how we wrote that rule using "x" and "n" ?</p>
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<div class="center80">
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<p align="center"><b>x<sub>n</sub></b> means "term number n", so term 3 is written <b>x<sub>3</sub></b></p>
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<p align="center"> </p>
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<p align="center">And we can calculate term 3 using:</p>
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<p class="large" align="center">x<sub>3</sub> = 3<sup>2</sup> = 9 </p>
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</div>
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<p> </p>
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<p>We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" <b>25</b> wherever <b>n</b> is.</p>
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<p class="large" align="center">x<sub>25</sub> = 25<sup>2</sup> = 625</p>
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<p> </p>
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<p>How about another example:</p>
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<div class="example">
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<h3>Example: 3, 5, 8, 13, 21, ?</h3>
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<p>After 3 and 5 all the rest are the <b>sum of the two numbers before</b>, </p>
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<p>That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the <a href="../numbers/fibonacci-sequence.html">Fibonacci Sequence</a>:</p>
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<p class="center">3, 5, 8, 13, 21, <b>34, 55, 89, ...</b></p>
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<p>Which has this Rule:</p>
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<p class="larger" align="center">Rule: <b>x<sub>n</sub> = x<sub>n-1</sub> + x<sub>n-2</sub></b> </p>
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</div>
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<p>Now what does <b>x<sub>n-1</sub></b> mean? It means "the previous term" as term number <b>n-1</b> is 1 less than term number <b>n</b>.</p>
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<p>And <b>x<sub>n-2</sub></b> means the term <i>before that one</i>. </p>
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<p>Let's try that Rule for the 6th term:</p>
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<p class="large" align="center">x<sub>6</sub> = x<sub>6-1</sub> + x<sub>6-2</sub></p>
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<p class="large" align="center">x<sub>6</sub> = x<sub>5</sub> + x<sub>4</sub></p>
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<p>So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:</p>
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<p class="large" align="center">x<sub>6</sub> = 21<sub></sub> + 13 = 34 </p>
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<h2>Many Rules</h2>
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<p>One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works. </p>
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<div class="example">
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<h3>What is the next number in the sequence 1, 2, 4, 7, ?</h3>
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<p>Here are three solutions (there can be more!):</p>
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<p><br>
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</p>
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<p class="large">Solution 1: Add 1, then add 2, 3, 4, ... </p>
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<p>So, 1+<b>1</b>=2, 2+<b>2</b>=4, 4+<b>3</b>=7, 7+<b>4</b>=11, etc...</p>
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<p><b>Rule: x<sub>n</sub> = n(n-1)/2 + 1</b></p>
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<p>Sequence: 1, 2, 4, 7, <b>11, 16, 22, ...</b></p>
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<p>(That rule looks a bit complicated, but it works)</p>
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<p> </p>
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<p class="large">Solution 2: After 1 and 2, add the two previous numbers, plus 1: </p>
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<p><b>Rule: x<sub>n</sub> = x<sub>n-1</sub> + x<sub>n-2</sub> + 1</b></p>
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<p>Sequence: 1, 2, 4, 7, <b>12, 20, 33, ...</b></p>
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<p> </p>
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<p class="large">Solution 3: After 1, 2 and 4, add the three previous numbers </p>
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<p><b>Rule: x<sub>n</sub> = x<sub>n-1</sub> + x<sub>n-2</sub> + x<sub>n-3</sub></b></p>
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<p>Sequence: 1, 2, 4, 7, <b>13, 24, 44, ...</b></p>
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</div>
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<p>So, we have three perfectly reasonable solutions, and they create totally different sequences.</p>
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<p align="center">Which is right? <b>They are all right.</b> </p>
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And there are other solutions ...
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<table border="0">
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<tbody>
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<tr>
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<td><img src="images/race.jpg" alt="race" height="72" width="150"></td>
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<td> </td>
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<td>
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<p>... it may be a list of the winners' numbers ... so the next number could be ... anything! </p></td>
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</tr>
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</tbody></table>
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<h2>Simplest Rule</h2>
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<p>When in doubt choose the <b>simplest rule</b> that makes sense, but also mention that there are other solutions.</p>
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<h2>Finding Differences</h2>
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<p>Sometimes it helps to find the <b>differences</b> between each pair of numbers ... this can often reveal an underlying pattern.</p>
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<p>Here is a simple case:</p>
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<p align="center"><img src="images/sequences-first-differences.gif" alt="sequences first differences" height="83" width="218"></p>
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<p>The differences are always 2, so we can guess that "2n" is part of the answer. </p>
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<p>Let us try <b>2n</b>:</p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<th width="150">n:</th>
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<th width="40">1</th>
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<th width="40">2</th>
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<th width="40">3</th>
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<th width="40">4</th>
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<th width="40">5</th>
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</tr>
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<tr align="center">
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<th width="150">Terms (x<sub>n</sub>):</th>
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<td>7</td>
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<td>9</td>
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<td>11</td>
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<td>13</td>
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<td>15</td>
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</tr>
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<tr align="center">
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<th width="150">2n:</th>
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<td>2</td>
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<td>4</td>
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<td>6</td>
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<td>8</td>
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<td>10</td>
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</tr>
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<tr align="center">
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<th width="150">Wrong by:</th>
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<td>5</td>
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<td>5</td>
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<td>5</td>
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<td>5</td>
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<td>5</td>
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</tr>
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</tbody></table>
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</div>
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<p>The last row shows that we are always wrong by 5, so just add 5 and we are done:</p>
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<p class="large" align="center">Rule: x<sub>n</sub> = 2n + 5</p>
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<p>OK, we could have worked out "2n+5" by just playing around with the numbers a bit, but we want a <b>systematic</b> way to do it, for when the sequences get more complicated.</p>
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<h2>Second Differences</h2>
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<p>In the sequence <b>{1, 2, 4, 7, 11, 16, 22, ...} </b>we need to find the differences ...</p>
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<div class="indent50px">
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<p> ... and then find the differences of <b>those</b> (called <i>second differences</i>), like this:</p>
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</div>
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<p align="center"><img src="images/sequences-second-differences.gif" alt="sequences second differences" height="141" width="296"></p>
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<p>The <b>second differences</b> in this case are 1. </p>
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<p class="larger" align="center">With second differences we multiply by <span class="intbl larger"><em>n<sup>2</sup></em><strong>2</strong></span></p>
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<p>In our case the difference is 1, so let us try just <span class="intbl larger"><em>n<sup>2</sup></em><strong>2</strong></span>:</p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<th width="150">n:</th>
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<th width="40">1</th>
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<th width="40">2</th>
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<th width="40">3</th>
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<th width="40">4</th>
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<th width="40">5</th>
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</tr>
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<tr align="center">
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<th width="150"><b>Terms (x<sub>n</sub>):</b></th>
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<td><b>1</b></td>
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<td><b>2</b></td>
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<td><b>4</b></td>
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<td><b>7</b></td>
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<td><b>11</b></td>
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</tr>
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<tr align="center">
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<th> </th>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<th><span class="intbl larger"><em>n<sup>2</sup></em><strong>2</strong></span>:</th>
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<td><b>0.5</b></td>
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<td><b>2</b></td>
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<td><b>4.5</b></td>
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<td><b>8</b></td>
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<td><b>12.5</b></td>
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</tr>
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<tr align="center">
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<th width="150">Wrong by:</th>
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<td>0.5</td>
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<td>0</td>
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<td>-0.5</td>
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<td>-1</td>
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<td>-1.5</td>
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</tr>
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</tbody></table>
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</div>
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<p>We are close, but seem to be drifting by 0.5, so let us try: <span class="larger"><span class="intbl"><em>n<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>n</em><strong>2</strong></span></span></p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<th><span class="intbl"><em>n<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>n</em><strong>2</strong></span></th>
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<td width="40"><b>0</b></td>
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<td width="40"><b>1</b></td>
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<td width="40"><b>3</b></td>
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<td width="40"><b>6</b></td>
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<td width="40"><b>10</b></td>
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</tr>
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<tr align="center">
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<th width="150">Wrong by:</th>
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<td>1</td>
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<td>1</td>
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<td>1</td>
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<td>1</td>
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<td>1</td>
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</tr>
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</tbody></table>
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</div>
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<p>Wrong by 1 now, so let us add 1:</p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<th><span class="intbl"><em>n<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>n</em><strong>2</strong></span> + 1</th>
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<td width="40"><b>1</b></td>
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<td width="40"><b>2</b></td>
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<td width="40"><b>4</b></td>
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<td width="40"><b>7</b></td>
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<td width="40"><b>11</b></td>
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</tr>
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<tr align="center">
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<th width="150">Wrong by:</th>
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<td>0</td>
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<td>0</td>
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<td>0</td>
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<td>0</td>
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<td>0</td>
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</tr>
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</tbody></table>
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</div>
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<p>We did it!</p>
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<p>The formula <b><span class="intbl"><em>n<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>n</em><strong>2</strong></span> + 1</b> can be simplified to <b>n(n-1)/2 + 1</b></p>
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<p>So by "trial-and-error" we discovered a rule that works:</p>
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<p align="center"><span class="larger">Rule: <b>x<sub>n</sub> = n(n-1)/2 + 1</b></span></p>
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<p>Sequence: 1, 2, 4, 7, 11, 16, 22, <b>29, 37, ...</b></p>
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<h2>Other Types of Sequences</h2>
|
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<p>Read <a href="sequences-series.html">Sequences and Series</a> to learn about:</p>
|
||
<ul>
|
||
<li><a href="sequences-sums-arithmetic.html">Arithmetic Sequences</a></li>
|
||
<li><a href="sequences-sums-geometric.html">Geometric Sequences</a></li>
|
||
<li><a href="../numbers/fibonacci-sequence.html">Fibonacci Sequence</a></li>
|
||
<li><a href="triangular-numbers.html">Triangular Sequence</a></li>
|
||
</ul>
|
||
<p>And there are also:</p>
|
||
<ul>
|
||
<li><a href="../prime_numbers.html">Prime Numbers</a></li>
|
||
<li><a href="../numbers/factorial.html">Factorial Numbers</a></li>
|
||
</ul>
|
||
<p>And many more! </p>
|
||
<p>In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just <a href="../contact.html">let me know</a>.</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
|
||
<script type="text/javascript">getQ(598, 3897, 3898, 1245, 599, 3013, 3899, 3014, 1246, 3900);</script>
|
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||
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="../numberpatterns.html">Number Patterns</a>
|
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<a href="sequences-series.html">Sequences and Series</a> </div>
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