32 lines
13 KiB
HTML
32 lines
13 KiB
HTML
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title: 數學歸納法 (Mathematical Induction)
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description:
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published: true
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date: 2026-02-11T17:20:14.172Z
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tags:
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editor: ckeditor
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dateCreated: 2026-02-11T17:14:35.544Z
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-->
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<h2><span style="font-family:Arial, Helvetica, sans-serif;">Fundamental Principle</span></h2>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Mathematical induction is a method used to prove that a statement holds true for all natural numbers k.</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Let P(n) be a statement defined for each positive integer n.</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Then P(n) will be true for all positive integers n if the following two conditions are satisfied:</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">(1) P(1) is true.</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">(2) P(k) is true for some integer k + 1 implies that P(k + 1) is true.</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Because all the other values can use the result from the (1) and (2),</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Therefore, you can conclude that all the conditions with integers n are true.</span></p>
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<p> </p>
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<h2><span style="font-family:Arial, Helvetica, sans-serif;">E1 Example</span></h2>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">Q Prove the following for all positive n:</span></p>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">A For n = 1, </span></p>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">The statement is true for n = 1.</span></p>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">For n = k + 1, Assume the statement is true for some integer k >= 1.</span></p>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<figure class="image"><img src="data:image/png;base64,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"></figure>
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<p><span style="font-family:Arial, Helvetica, sans-serif;">The statement is true for n = k + 1. By the principle of mathematical induction, the statement is true for all positive integers n.</span></p>
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