From 174229b65bc22f7dcedd0e474748a5de8c5424c7 Mon Sep 17 00:00:00 2001
From: phidias <phidias@aperx.org>
Date: Wed, 11 Feb 2026 17:20:56 +0000
Subject: [PATCH] docs: update education/mathematics/mathematical-induction

---
 education/mathematics/mathematical-induction.html | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/education/mathematics/mathematical-induction.html b/education/mathematics/mathematical-induction.html
index d8fa069..6259e66 100644
--- a/education/mathematics/mathematical-induction.html
+++ b/education/mathematics/mathematical-induction.html
@@ -2,7 +2,7 @@
 title: 數學歸納法 (Mathematical Induction)
 description: 
 published: true
-date: 2026-02-11T17:20:14.172Z
+date: 2026-02-11T17:20:55.457Z
 tags: 
 editor: ckeditor
 dateCreated: 2026-02-11T17:14:35.544Z
@@ -29,3 +29,11 @@ dateCreated: 2026-02-11T17:14:35.544Z
 <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;">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>
+<p>&nbsp;</p>
+<h2><span style="font-family:Arial, Helvetica, sans-serif;">Variation</span></h2>
+<p><span style="font-family:Arial, Helvetica, sans-serif;">There were cases where proof for each positive integer was needed, instead of all the integers.</span></p>
+<p>&nbsp;</p>
+<h3><span style="font-family:Arial, Helvetica, sans-serif;">For instance:</span></h3>
+<p><span style="font-family:Arial, Helvetica, sans-serif;">When n is in a range.</span></p>
+<p><span style="font-family:Arial, Helvetica, sans-serif;">For all positive odd integers n.</span></p>
+<p><span style="font-family:Arial, Helvetica, sans-serif;">For all positive real numbers.</span></p>