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
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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>
+<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>