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131 lines
5.3 KiB
HTML
131 lines
5.3 KiB
HTML
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<title>Tetrahedral Number Sequence</title>
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<h1 align="center">Tetrahedral Number Sequence</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/tetrahedral-number.jpg" width="200" height="230" alt="tetrahedral number" /></p>
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<p>This is the Tetrahedral Number Sequence:</p>
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<p class="center"> <b>1, 4, 10, 20, 35, ...</b></p>
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<p>We can understand it better when we think of a stack of marbles in the shape of a <a href="geometry/tetrahedron.html">Tetrahedron</a>. </p>
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<p>Just count how many marbles are needed for a stack of a certain height.</p>
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<ul>
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<li>For height=1 we only need <b>1</b> marble</li>
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<li>For height=2, we need <b>4</b> marbles (1 at the top and 3 below)</li>
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<li>For height=3 we need <b>10</b> marbles.</li>
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<li>For height=4 we need <b>20</b> marbles.</li>
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<li>For height=5 we need <b>35</b> marbles.</li>
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<li>How many for height=6 ... ?</li>
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</ul>
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<h2>Triangular and Tetrahedral Numbers</h2>
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<p>Each <b>layer</b> in the tetrahedron of marbles is actually part of the <a href="algebra/triangular-numbers.html">Triangular Number Sequence</a> (1, 3, 6, etc). And both the triangular numbers and the tetrahedral numbers are on <a href="pascals-triangle.html">Pascal's Triangle</a>.</p>
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<p>This table shows the values for the first few layers:</p>
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<div class="beach">
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<table border="0" align="center">
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<tr align="center">
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<th>n</th>
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<th><b>Triangular Number</b></th>
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<th><b>Tetrahedral Number</b></th>
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</tr>
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<tr align="center">
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<td>(Height)</td>
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<td>(Marbles in Layer)</td>
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<td>(Total Marbles)</td>
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</tr>
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<tr align="center">
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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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<tr align="center">
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<td>2</td>
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<td>3</td>
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<td>4</td>
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</tr>
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<tr align="center">
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<td>3</td>
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<td>6</td>
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<td>10</td>
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</tr>
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<tr align="center">
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<td>4</td>
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<td>10</td>
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<td>20</td>
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</tr>
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<tr align="center">
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<td>5</td>
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<td>15</td>
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<td>35</td>
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</tr>
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<tr align="center">
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<td>6</td>
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<td>21</td>
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<td>56</td>
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</tr>
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</table>
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<p>Looking at the numbers we see something interesting: when we take any number and add the number before it and to the left we get the next number in the sequence. (For example 4+6=10).</p>
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<p> </p>
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</div>
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<div class="related"> <a href="geometry/tetrahedron.html">Tetrahedron</a> <a href="algebra/triangular-numbers.html">Triangular Number Sequence</a> <a href="pascals-triangle.html">Pascal's Triangle</a> </div>
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