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535 lines
24 KiB
HTML
535 lines
24 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Fibonacci Sequence</h1>
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<p>The Fibonacci Sequence is the series of numbers:</p>
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<p class="center"><span class="large">0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...</span></p>
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<p>The next number is found by adding up the two numbers before it:</p>
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<ul>
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<li>the 2 is found by adding the two numbers before it (1+1),</li>
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<li>the 3 is found by adding the two numbers before it (1+2),</li>
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<li>the 5 is (2+3),</li>
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<li>and so on!</li>
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</ul>
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<div class="example">
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<h3>Example: the next number in the sequence above is 21+34 = <b>55</b></h3>
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</div>
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<p>It is that simple!</p>
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<p>Here is a longer list:</p>
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<p>0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...</p>
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<p class="center"><i>Can you figure out the next few numbers? </i></p>
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<h2>Makes A Spiral</h2>
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<p>When we make squares with those widths, we get a nice spiral:</p>
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<p class="center"><img src="images/fibonacci-spiral.svg" alt="Fibonacci Spiral" style="max-width:100%"></p>
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<p class="center">Do you see how the squares fit neatly together?<br>
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For example 5 and 8 make 13, 8 and 13 make 21, and so on.</p>
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<p class="center"><img src="images/sunflower.jpg" alt="sunflower" height="283" width="400"><br>
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This spiral is found in nature!<br>
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See: <a href="nature-golden-ratio-fibonacci.html">Nature, The Golden Ratio,
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and Fibonacci</a></p>
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<h2>The Rule</h2>
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<p>The Fibonacci Sequence can be written as a "Rule" (see <a href="../algebra/sequences-series.html">Sequences and Series</a>).</p>
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<p>First, the terms are numbered from 0 onwards like this:</p>
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<div class="beach">
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<table align="center" cellspacing="0" cellpadding="0">
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<tbody>
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<tr>
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<td align="center" width="60"><i>n =</i></td>
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<td align="center" width="35"><i>0</i></td>
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<td align="center" width="35"><i>1</i></td>
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<td align="center" width="35"><i>2</i></td>
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<td align="center" width="35"><i>3</i></td>
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<td align="center" width="35"><i>4</i></td>
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<td align="center" width="35"><i>5</i></td>
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<td align="center" width="35"><i>6</i></td>
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<td align="center" width="35"><i>7</i></td>
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<td align="center" width="35"><i>8</i></td>
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<td align="center" width="35"><i>9</i></td>
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<td align="center" width="35"><i>10</i></td>
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<td align="center" width="35"><i>11</i></td>
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<td align="center" width="35"><i>12</i></td>
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<td align="center" width="35"><i>13</i></td>
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<td align="center" width="35"><i>14</i></td>
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<td align="center" width="35"><i>...</i></td>
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</tr>
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<tr class="large">
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<td align="center" width="60">x<sub>n</sub> =</td>
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<td align="center" width="35">0</td>
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<td align="center" width="35">1</td>
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<td align="center" width="35">1</td>
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<td align="center" width="35">2</td>
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<td align="center" width="35">3</td>
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<td align="center" width="35">5</td>
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<td align="center" width="35">8</td>
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<td align="center" width="35">13</td>
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<td align="center" width="35">21</td>
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<td align="center" width="35">34</td>
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<td align="center" width="35">55</td>
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<td align="center" width="35">89</td>
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<td align="center" width="35">144</td>
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<td align="center" width="35">233</td>
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<td align="center" width="35">377</td>
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<td align="center" width="35">...</td>
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</tr>
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</tbody></table>
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</div>
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<p>So term number 6 is called <span class="larger">x<sub>6</sub></span> (which equals 8).</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>
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<p>Example: the <b>8th</b> term is<br>
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the <b>7th</b> term plus the <b>6th</b> term:</p>
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<p class="center"><br>
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<span class="larger">x<sub>8</sub> = x<sub>7</sub> + x<sub>6</sub></span></p>
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</td>
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<td><img src="images/fibonacci-rule.gif" alt="fibonacci rule x_8 = x_7 + x_6" height="136" width="269"></td>
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</tr>
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</tbody></table>
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</div>
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<p>So we can write the rule:</p>
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<p class="center larger">The Rule is <b> x<sub>n</sub> = x<sub>n−1</sub> + x<sub>n−2</sub></b></p>
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<p>where:</p>
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<ul>
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<li><b>x<sub>n</sub></b> is term number "n"</li>
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<li><b>x<sub>n−1</sub></b> is the previous term (n−1)</li>
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<li><b>x<sub>n−2</sub></b> is the term before that (n−2)</li>
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</ul>
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<div class="example">
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<h3>Example: term 9 is calculated like this:</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="larger">x<sub>9</sub></span></span><span class="right"><span class="larger">= x<sub>9−1</sub> + x<sub>9−2</sub></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= x<sub>8</sub> + x<sub>7</sub></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= 21 + 13 </span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= 34</span></span></div>
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</div>
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</div>
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<h2>Golden Ratio</h2>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/golden-rectangle.svg" alt="golden rectangle"></p>
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<p>And here is a surprise. When we take any two successive <i>(one after the other)</i> Fibonacci Numbers, their ratio is very close to the <a href="golden-ratio.html">Golden Ratio</a> "<b>φ</b>" which is approximately <b>1.618034...</b></p>
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<p>In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th width="50">
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<div align="right">A </div> </th>
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<th width="50">
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<div align="right">B </div> </th>
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<th width="20"> </th>
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<th width="100">
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<div align="left">B / A</div> </th>
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</tr>
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<tr>
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<td width="50">
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<div align="right">2</div> </td>
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<td width="50">
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<div align="right">3</div> </td>
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<td width="20"> </td>
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<td width="100">1.5</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">3</div> </td>
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<td width="50">
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<div align="right">5</div> </td>
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<td width="20"> </td>
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<td width="100">1.666666666...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">5</div> </td>
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<td width="50">
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<div align="right">8</div> </td>
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<td width="20"> </td>
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<td width="100">1.6</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">8</div> </td>
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<td width="50">
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<div align="right">13</div> </td>
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<td width="20"> </td>
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<td width="100">1.625</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div> </td>
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<td height="14" width="50">
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<div align="right">...</div> </td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">144</div> </td>
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<td width="50">
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<div align="right">233</div> </td>
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<td width="20"> </td>
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<td width="100">1.618055556...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">233</div> </td>
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<td width="50">
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<div align="right">377</div> </td>
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<td width="20"> </td>
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<td width="100">1.618025751...</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div> </td>
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<td height="14" width="50">
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<div align="right">...</div> </td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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</tbody></table>
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<p>We don't have to start with <b>2 and 3</b>, here I randomly chose <b>192 and 16</b> (and got the sequence <i>192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...</i>):</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th width="50"> <div align="right">A </div></th>
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<th width="50"> <div align="right">B </div></th>
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<th width="20"> </th>
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<th width="100"> <div align="left">B / A</div></th>
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</tr>
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<tr>
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<td width="50">
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<div align="right"><b>192</b></div></td>
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<td width="50">
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<div align="right"><b>16</b></div></td>
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<td width="20"> </td>
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<td width="100">0.08333333...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">16</div></td>
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<td width="50">
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<div align="right">208</div></td>
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<td width="20"> </td>
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<td width="100">13</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">208</div></td>
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<td width="50">
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<div align="right">224</div></td>
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<td width="20"> </td>
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<td width="100">1.07692308...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">224</div></td>
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<td width="50">
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<div align="right">432</div></td>
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<td width="20"> </td>
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<td width="100">1.92857143...</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div></td>
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<td height="14" width="50">
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<div align="right">...</div></td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">7408</div></td>
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<td width="50">
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<div align="right">11984</div></td>
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<td width="20"> </td>
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<td width="100">1.61771058...</td>
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</tr>
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<tr>
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<td width="50">
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<div align="right">11984</div></td>
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<td width="50">
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<div align="right">19392</div></td>
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<td width="20"> </td>
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<td width="100">1.61815754...</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div></td>
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<td height="14" width="50">
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<div align="right">...</div></td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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</tbody></table>
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<p>It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!</p>
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<h2>Using The Golden Ratio to Calculate Fibonacci Numbers</h2>
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<p>And even more surprising is that we can <b>calculate any Fibonacci Number</b> using the Golden Ratio:</p>
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<p class="center large">x<sub>n</sub> = <span class="intbl"><em>φ<sup>n</sup> − (1−φ)<sup>n</sup></em><strong>√5</strong></span></p>
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|
||
<p>The answer comes out <b>as a whole number</b>, exactly equal to the addition of the previous two terms.</p>
|
||
<div class="example">
|
||
<h3>Example: x<sub>6</sub></h3>
|
||
<p class="center larger">x<sub>6</sub> = <span class="intbl"><em>(1.618034...)<sup>6</sup> − (1−1.618034...)<sup>6</sup></em><strong>√5</strong></span></p>
|
||
|
||
<p>When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer <b>8.00000033</b> , a more accurate calculation would be closer to 8.</p>
|
||
<p>Try n=12 and see what you get.</p>
|
||
</div>
|
||
|
||
<p>You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):</p>
|
||
<div class="example">
|
||
<h3>Example: 8 × φ = 8 × 1.618034... = 12.94427... = 13 (rounded)</h3>
|
||
</div>
|
||
<h2>Some Interesting Things</h2>
|
||
<div class="fun">
|
||
<p>Here is the Fibonacci sequence again:</p>
|
||
<div class="beach">
|
||
<table align="center" cellspacing="0" cellpadding="0">
|
||
<tbody>
|
||
<tr>
|
||
<td align="center" width="60"><i>n =</i></td>
|
||
<td align="center" width="35"><i>0</i></td>
|
||
<td align="center" width="35"><i>1</i></td>
|
||
<td align="center" width="35"><i>2</i></td>
|
||
<td align="center" width="35"><i>3</i></td>
|
||
<td align="center" width="35"><i>4</i></td>
|
||
<td align="center" width="35"><i>5</i></td>
|
||
<td align="center" width="35"><i>6</i></td>
|
||
<td align="center" width="35"><i>7</i></td>
|
||
<td align="center" width="35"><i>8</i></td>
|
||
<td align="center" width="35"><i>9</i></td>
|
||
<td align="center" width="35"><i>10</i></td>
|
||
<td align="center" width="35"><i>11</i></td>
|
||
<td align="center" width="35"><i>12</i></td>
|
||
<td align="center" width="35"><i>13</i></td>
|
||
<td align="center" width="35"><i>14</i></td>
|
||
<td align="center" width="35"><i>15</i></td>
|
||
<td align="center" width="35"><i>...</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center" width="60">x<sub>n</sub> =</td>
|
||
<td align="center" width="35">0</td>
|
||
<td align="center" width="35">1</td>
|
||
<td align="center" width="35">1</td>
|
||
<td align="center" width="35">2</td>
|
||
<td align="center" width="35">3</td>
|
||
<td align="center" width="35">5</td>
|
||
<td align="center" width="35">8</td>
|
||
<td align="center" width="35">13</td>
|
||
<td align="center" width="35">21</td>
|
||
<td align="center" width="35">34</td>
|
||
<td align="center" width="35">55</td>
|
||
<td align="center" width="35">89</td>
|
||
<td align="center" width="35">144</td>
|
||
<td align="center" width="35">233</td>
|
||
<td align="center" width="35">377</td>
|
||
<td align="center" width="35">610</td>
|
||
<td align="center" width="35">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>There is an interesting pattern:</p>
|
||
<ul>
|
||
<li>Look at the number <b>x<sub>3</sub> = 2</b>. Every <b>3</b>rd number is a multiple of <b>2</b> (2, 8, 34, 144, 610, ...)</li>
|
||
<li>Look at the number <b>x<sub>4</sub> = 3</b>. Every <b>4</b>th number is a multiple of <b>3</b> (3, 21, 144, ...)</li>
|
||
<li>Look at the number <b>x<sub>5</sub> = 5</b>. Every <b>5</b>th number is a multiple of <b>5</b> (5, 55, 610, ...)</li>
|
||
</ul>
|
||
<p>And so on (every <b>n</b>th number is a multiple of<b> x<sub>n</sub></b>).</p>
|
||
</div><p> </p>
|
||
<div class="fun">
|
||
<h3>1/89 = 0.011235955056179775...</h3>
|
||
<p>Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?</p>
|
||
<p>In a way they <b>all</b> are, except multiple digit numbers (13, 21, etc) <b>overlap</b>, like this:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>0.0</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.01</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.001</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.0002</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.00003</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.000005</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.0000008</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.00000013</td>
|
||
</tr>
|
||
<tr>
|
||
<td>0.000000021</td>
|
||
</tr>
|
||
<tr>
|
||
<td> ... etc ...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border-top: 2px solid black"><br></td>
|
||
</tr>
|
||
<tr>
|
||
<td><b>0.011235955056179775...</b> = 1/89</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<h2>Terms Below Zero</h2>
|
||
<p>The sequence works below zero also, like this:</p>
|
||
<div class="beach">
|
||
<table align="center" cellspacing="0" cellpadding="0">
|
||
<tbody>
|
||
<tr>
|
||
<td align="center" width="60"><i>n =</i></td>
|
||
<td align="center" width="35"><i>...</i></td>
|
||
<td align="center" width="35"><i>−6</i></td>
|
||
<td align="center" width="35"><i>−5</i></td>
|
||
<td align="center" width="35"><i>−4</i></td>
|
||
<td align="center" width="35"><i>−3</i></td>
|
||
<td align="center" width="35"><i>−2</i></td>
|
||
<td align="center" width="35"><i>−1</i></td>
|
||
<td align="center" width="35"><i><b>0</b></i></td>
|
||
<td align="center" width="35"><i>1</i></td>
|
||
<td align="center" width="35"><i>2</i></td>
|
||
<td align="center" width="35"><i>3</i></td>
|
||
<td align="center" width="35"><i>4</i></td>
|
||
<td align="center" width="35"><i>5</i></td>
|
||
<td align="center" width="35"><i>6</i></td>
|
||
<td align="center" width="35"><i>...</i></td>
|
||
</tr>
|
||
<tr class="large">
|
||
<td align="center" width="60">x<sub>n</sub> =</td>
|
||
<td align="center" width="35">...</td>
|
||
<td align="center" width="35">−8</td>
|
||
<td align="center" width="35">5</td>
|
||
<td align="center" width="35">−3</td>
|
||
<td align="center" width="35">2</td>
|
||
<td align="center" width="35">−1</td>
|
||
<td align="center" width="35">1</td>
|
||
<td align="center" width="35">0</td>
|
||
<td align="center" width="35">1</td>
|
||
<td align="center" width="35">1</td>
|
||
<td align="center" width="35">2</td>
|
||
<td align="center" width="35">3</td>
|
||
<td align="center" width="35">5</td>
|
||
<td align="center" width="35">8</td>
|
||
<td align="center" width="35">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p class="center"><i>(Prove to yourself that each number is found by adding up the two numbers before it!)</i></p>
|
||
<p>In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a <span class="mono">+-+-</span> ... pattern. It can be written like this:</p>
|
||
<p class="center"><span class="large"><b>x</b><sub>−n</sub> = (−1)<sup>n+1</sup> <b>x</b><sub>n</sub></span></p>
|
||
<p>Which says that term "−n" is equal to <span class="large">(−1)<sup>n+1</sup></span> times term "n", and the value <span class="large">(−1)<sup>n+1</sup></span> neatly makes the correct +1, −1, +1, −1, ... pattern.</p>
|
||
<h2>History</h2>
|
||
<p>Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!</p>
|
||
|
||
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/fibonacci.jpg" alt="fibonacci portrait" height="163" width="134"></p>
|
||
<h2>About Fibonacci The Man</h2>
|
||
<p>His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.</p>
|
||
<p>"Fibonacci" was his nickname, which roughly means "Son of Bonacci".</p>
|
||
<p>As well as being famous for the Fibonacci Sequence, he helped spread <a href="../place-value.html">Hindu-Arabic Numerals</a> (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of <a href="../roman-numerals.html">Roman Numerals</a> (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.</p>
|
||
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../images/balloons.gif" alt="balloons" height="110" width="100"></p>
|
||
<h2>Fibonacci Day</h2>
|
||
<p>Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!</p>
|
||
<p> </p>
|
||
|
||
<div class="questions">
|
||
<script>getQ(1738, 8338, 8339, 8340, 8341, 8343, 3012, 3071, 8841, 8367);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="golden-ratio.html">Golden Ratio</a>
|
||
<a href="nature-golden-ratio-fibonacci.html">Nature, Golden Ratio and Fibonacci Numbers</a>
|
||
<a href="../numberpatterns.html">Number Patterns</a>
|
||
</div>
|
||
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