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142 lines
5.5 KiB
HTML
142 lines
5.5 KiB
HTML
<!doctype html>
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<html lang="en">
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<!-- Mirrored from www.mathsisfun.com/algebra/triangular-numbers.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:03:21 GMT -->
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<head>
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<meta charset="UTF-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Triangular Number Sequence</title>
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<meta name="description" content="The Triangular Number Sequence comes from a pattern of dots that form a triangle" />
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Triangular Number Sequence</h1>
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<p>This is the Triangular Number Sequence:</p>
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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, 3, 6, 10, 15, 21, 28, 36, 45</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>It is simply the number of dots in each <b>triangular pattern</b>:</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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<p class="center">By adding another row of dots and counting all the dots we can<br>
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find
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the next number of the sequence.</p>
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<ul>
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<li>The first triangle has just one dot.</li>
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<li>The second triangle has another row with 2 extra dots, making 1<b> + 2</b> = 3</li>
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<li>The third triangle has another row with 3 extra dots, making 1 + 2 <b>+ 3</b> = 6</li>
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<li>The fourth has 1 + 2 + 3 <b>+ 4</b> = 10</li>
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<li>etc!</li>
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</ul>
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<p><i>How may dots in the 60th triangle?</i></p>
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<h2>A Rule</h2>
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<p>We can make a "Rule" so we can calculate any triangular number.</p>
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<p>First, rearrange the dots like this:</p>
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<p class="center"><img src="../numbers/images/triangular-number-2.svg" alt="triangular numbers 1 to 5" style="max-width:100%" height="102" width="461" ></p>
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<p>Then double the number of dots, and form them into a rectangle:</p>
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<p class="center"><img src="../numbers/images/triangular-number-3.svg" alt="triangular numbers when doubled become n by n+1 rectangles" style="max-width:100%" height="101" width="471" ></p>
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<p><i>Now</i> it is easy to work out how many dots: just multiply <b>n</b> by <b>n+1</b></p>
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<p class="center larger">Dots in rectangle = n(n+1)</p>
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<p>But remember we doubled the number of dots, so</p>
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<p class="center larger">Dots in triangle = n(n+1)/2</p>
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<p>We can use <span class="large">x<sub>n</sub></span> to mean "dots in triangle n", so we get the rule:</p>
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<p class="center"><span class="large">Rule: x<sub>n</sub> = n(n+1)/2</span></p>
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<div class="example">
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<p>Example: the <b>5th</b> Triangular Number is</p>
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<p class="center">x<sub>5</sub> = 5(5+1)/2 = <b>15</b></p>
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</div>
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<div class="example">
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<p>Example: the <b>60th</b> is</p>
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<p class="center">x<sub>60</sub> = 60(60+1)/2 = <b>1830</b></p>
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<p class="center">Wasn't it much easier to use the formula than to add up all those dots?</p>
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</div>
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<div class="example"><p style="float:right; margin: 0 0 5px 10px;"><img src="images/log-stack.jpg" alt="log stack" height="317" width="250" ></p>
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<h3>Example: You are stacking logs.</h3>
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<p>There is enough ground for you to lay 22 logs side-by-side.</p>
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<p>How many logs can you fit in the stack?</p>
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<p class="center">x<sub>22</sub> = 22(22+1)/2 = <b>253</b></p>
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<p> </p>
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<p>The stack may be dangerously high, but you can fit 253 logs in it!</p>
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</div>
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<p> </p>
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<div class="questions"></div>
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<div class="related">
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<a href="sequences-series.html">Number Sequences</a>
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<a href="index.html">Algebra Index</a>
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<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
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