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docs: update education/mathematics/mathematical-induction

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2026-02-11 17:21:19 +00:00
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title: 數學歸納法 (Mathematical Induction) title: 數學歸納法 (Mathematical Induction)
description: description:
published: true published: true
date: 2026-02-11T17:20:55.457Z date: 2026-02-11T17:21:19.387Z
tags: tags:
editor: ckeditor editor: ckeditor
dateCreated: 2026-02-11T17:14:35.544Z dateCreated: 2026-02-11T17:14:35.544Z
@@ -17,7 +17,7 @@ dateCreated: 2026-02-11T17:14:35.544Z
<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> <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>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Therefore, you can conclude that all the conditions with integers n are true.</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">Therefore, you can conclude that all the conditions with integers n are true.</span></p>
<p>&nbsp;</p> <p>&nbsp;</p>
<h2><span style="font-family:Arial, Helvetica, sans-serif;">E1 Example</span></h2> <h3><span style="font-family:Arial, Helvetica, sans-serif;">E1 Example</span></h3>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Q Prove the following for all positive n:</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">Q Prove the following for all positive n:</span></p>
<figure class="image"><img src="data:image/png;base64,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"></figure> <figure class="image"><img src="data:image/png;base64,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"></figure>
<p><span style="font-family:Arial, Helvetica, sans-serif;">A For n = 1,&nbsp;</span></p> <p><span style="font-family:Arial, Helvetica, sans-serif;">A For n = 1,&nbsp;</span></p>