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344 lines
11 KiB
HTML
344 lines
11 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/numberpatterns.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:01:47 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Number Sequences - Square, Cube and Fibonacci</title>
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<div id="hdr"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="index.html"><img src="images/style/logo.svg" alt="Math is Fun"></a></div>
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<!-- #BeginEditable "Body" -->
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<center>
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<h1>Common Number Patterns</h1>
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</center>
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<h3 align="center">Numbers can have interesting patterns.<br>
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Here we list the most common patterns and how they are made.</h3>
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<div class="video">GijrTfz0tIU</div>
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<h2 id="titleas">Arithmetic Sequences</h2>
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<p>An <a href="algebra/sequences-sums-arithmetic.html">Arithmetic Sequence</a> is made by <b>adding</b> the same value each time.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">1, 4, 7, 10, 13, 16, 19, 22, 25, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence has a difference of 3 between each number.<br>
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The pattern is continued by <b>adding 3</b> to the last number each time, like this:</p>
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<p class="center"><img src="algebra/images/arithmetic-sequence-1-3.svg" alt="arithmetic sequence 1, 4, 7, 10," height="80" width="360"></p>
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</div>
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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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<div class="example2">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">3, 8, 13, 18, 23, 28, 33, 38, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence has a difference of 5 between each number.<br>
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The pattern is continued by <b>adding 5</b> to the last number each time, like this:</p>
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<p class="center"><img src="algebra/images/arithmetic-sequence-3-5.svg" alt="arithmetic sequence 3, 8, 13, 18" height="80" width="360"></p>
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</div>
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<p>The value added each time is called the <b>"common difference"</b></p>
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<p>What is the common difference in this example?</p>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">19, 27, 35, 43, ...</td>
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</tr>
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</tbody></table>
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</div>
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<div class="center"><br>
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Answer: <span class="hide">The common difference is <b>8</b></span> </div>
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<p><a id="qa"></a>The common difference could also be negative:</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">25, 23, 21, 19, 17, 15, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This common difference is <b>−2</b><br>
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The pattern is continued by <b>subtracting 2</b> each time, like this:</p>
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<p class="center"><img src="algebra/images/arithmetic-sequence-25-2.svg" alt="arithmetic sequence 25, 23, 21,..." height="74" width="198"></p>
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</div>
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<div class="questions">1739, 1740, 2511, 9771, 9772</div>
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<p> </p>
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<div class="video">nwazO55Bkzo</div>
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<h2 id="titlegs">Geometric Sequences</h2>
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<p>A <a href="algebra/sequences-sums-geometric.html">Geometric Sequence</a> is made by <b>multiplying</b> by the same value each time.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">1, 3, 9, 27, 81,243, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence has a factor of 3 between each number.<br>
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The pattern is continued by <b>multiplying by 3</b> each
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time, like this:</p>
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<p class="center"><img src="algebra/images/geometric-sequence-3.svg" alt="geometric sequence 1, 3, 9," height="83" width="400"></p>
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</div>
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<p>What we multiply by each time is called the "<b>common ratio</b>".</p>
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<p>In the previous example the common ratio was <b>3</b>:</p>
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<p class="center"><img src="algebra/images/geometric-sequence-3a.svg" alt="geometric sequence 1, 3, 9, common ratio 3" style="max-width:100%" height="116" width="406"></p>
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<p>We can start with any number:</p>
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<div class="example">
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<h3>Example: Common Ratio of 3, But Starting at 2</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">2, 6, 18, 54,162,486, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence <b>also</b> has a common ratio of 3, but it starts with <b>2</b>.</p>
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<p class="center"><img src="algebra/images/geometric-sequence-3b.svg" alt="geometric sequence 2, 6, 18" height="80" width="442"><br></p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large"> 1, 2, 4, 8, 16, 32, 64,128,256, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence starts at 1 and has a <b>common ratio of 2</b>.<br>
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The pattern is continued by <b>multiplying by 2</b> each
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time, like this:</p>
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<p class="center"><img src="algebra/images/geometric-sequence-2.svg" alt="geometric sequence 1, 2, 4, 8, 16," height="85" width="395"></p>
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</div>
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<p><a id="qg"></a>The common ratio can be less than 1:</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">10, 5, 2.5, 1.25, 0.625, 0.3125, ...</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>This sequence starts at 10 and has a common ratio of 0.5 (a half).<br>
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The pattern is continued by <b>multiplying by 0.5</b> each
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time.</p>
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</div>
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<p>But the common ratio can't be 0, as we get a sequence like 1, 0, 0, 0, 0, 0, 0, ...</p>
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<div class="questions">658,796, 1741, 10006, 10007</div>
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<h2>Special Sequences</h2>
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<p>There are also <i><b>many</b></i> special sequences, here are some of the most common:</p>
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<h2><a name="triangular" id="triangular"></a>Triangular Numbers</h2>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">1, 3, 6, 10, 15, 21, 28, 36, 45, ...</td>
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</tr>
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</tbody></table>
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</div>
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<p>This <a href="algebra/triangular-numbers.html">Triangular Number Sequence</a> is generated from a pattern of dots that form a
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triangle.</p>
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<p>By adding another row of dots and counting all the dots we can find
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the next number of the sequence:</p>
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<p class="center"><img src="numbers/images/triangular-number-dots.svg" alt="triangular numbers" style="max-width:100%" height="177" width="473"></p>
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<h2>Square Numbers</h2>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...</td>
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</tr>
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</tbody></table>
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</div>
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<p>They are the <a href="square-root.html">squares</a> of whole numbers:</p>
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<p>0 (=0×0)<br>
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1 (=1×1)<br>
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4 (=2×2)<br>
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9 (=3×3)<br>
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16 (=4×4)<br>
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etc...</p>
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<h2>Cube Numbers</h2>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large"> 1, 8, 27, 64,125,216,343,512,729, ...</td>
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</tr>
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</tbody></table>
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</div>
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<p>They are the <a href="numbers/cube-root.html">cubes</a> of the counting numbers (they start at 1):</p>
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<p>1 (=1×1×1)<br>
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8 (=2×2×2)<br>
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27 (=3×3×3)<br>
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64 (=4×4×4)<br>
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etc...</p>
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<h2>Fibonacci Numbers</h2>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td class="large">0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...</td>
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</tr>
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</tbody></table>
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</div>
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<p>The <a href="numbers/fibonacci-sequence.html">Fibonacci Sequence</a> is found by adding the two numbers before it together.<br>
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The 2 is found by adding the two numbers before it (1+1)<br>
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The 21 is found by adding the two numbers before it (8+13)<br>
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The next number in the sequence above would be 55 (21+34)<br>
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<br>
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Can you figure out the <b>next</b> few numbers?</p>
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<h2>Other Sequences</h2>
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<p>There are lots more! You might even <a href="your-own-number-patterns.html">think of your own</a> ...</p>
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<p> </p>
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<div class="questions">1736, 1737, 3860, 3861, 3862, 1735, 1738</div>
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<div class="related">
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<a href="your-own-number-patterns.html">Your Own Number Patterns</a>
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<a href="pascals-triangle.html">Pascal's Triangle</a>
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<a href="algebra/sequences-series.html">Sequences and Series</a>
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<a href="algebra/sequence-diff-tool.html">Sequence Difference Tool</a>
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<a href="numbers/fibonacci-sequence.html">Fibonacci Sequence</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
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