7.4 - Mathematical Induction

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<?xml version="1.0" encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>7.4 - Mathematical Induction</title> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> </head> <body> <h1>7.4 - Mathematical Induction</h1> <h2>The need for proof</h2> <p>Most people today are lazy. We watch way too much television and are content to accept things as true without question. </p> <p>If we see something that works a few times in a row, we're convinced that it works forever.</p> <h3>Regions of a Circle</h3> <p>Consider a circle with n points on it. How many regions will the circle be divided into if each pair of points is connected with a chord? </p> <table border="border" cellpadding="5"> <tr> <td valign="top"><img src="circle2.gif" alt="circle with 2 connected points" width="125" height="124" /></td> <td valign="top"><img src="circle3.gif" alt="circle with 3 connected points" width="125" height="124" /></td> <td valign="top"><img src="circle4.gif" alt="circle with 4 connected points" width="125" height="124" /></td> <td valign="top"><img src="circle5.gif" alt="circle with 5 connected points" width="125" height="124" /></td> </tr> <tr> <td valign="top">2 points<br /> 2 regions = 2<sup>1</sup></td> <td valign="top">3 points<br /> 4 regions = 2<sup>2</sup></td> <td valign="top">4 points<br /> 8 regions = 2<sup>3</sup></td> <td valign="top">5 points<br /> 16 regions = 2<sup>4</sup></td> </tr></table> <p>At this point, probably everyone would be convinced that with 6 points there would be 32 regions, but it's not proved, it's just conjectured. The conjecture is that the number of regions when n points are connected is 2<sup>n -1</sup>. </p> <p>Will finding the number of regions when there are six points on the circle prove the conjecture? No. If there are indeed 32 regions, all you have done is shown another example to support your conjecture. If there aren't 32 regions, then you have proved the conjecture wrong. In fact, if you go ahead and try the circle with six points on it, you'll find
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out that there aren't 32 regions.</p> <p><strong>You can never prove a conjecture is true by example. </strong></p> <p><strong>You can prove a conjecture is false by finding a counter- example.</strong> </p> <p>To prove a conjecture is true, you need some more formal methods of proof. One of these methods is the principle of mathematical induction.</p> <h2>Principle of Mathematical Induction (English)</h2> <ol> <li>Show something works the first time.</li> <li>Assume that it works for this time,</li> <li>Show it will work for the next time.</li> <li>Conclusion, it works all the time</li> </ol> <h2>Principle of Mathematical Induction (Mathematics)</h2> <ol> <li>Show true for n = 1</li> <li>Assume true for n = k</li> <li>Show true for n = k + 1</li> <li>Conclusion: Statement is true for all n >= 1</li>
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7.4 - Mathematical Induction - &lt;?xml...

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