From ed4a54ec0f98c6eb6cf951be8074130285c69451 Mon Sep 17 00:00:00 2001 From: phidias Date: Wed, 11 Feb 2026 17:20:14 +0000 Subject: [PATCH] docs: update education/mathematics/mathematical-induction --- .../mathematics/mathematical-induction.html | 31 +++++++++++++------ 1 file changed, 22 insertions(+), 9 deletions(-) diff --git a/education/mathematics/mathematical-induction.html b/education/mathematics/mathematical-induction.html index e5b2fa4..d8fa069 100644 --- a/education/mathematics/mathematical-induction.html +++ b/education/mathematics/mathematical-induction.html @@ -2,17 +2,30 @@ title: 數學歸納法 (Mathematical Induction) description: published: true -date: 2026-02-11T17:16:01.234Z +date: 2026-02-11T17:20:14.172Z tags: editor: ckeditor dateCreated: 2026-02-11T17:14:35.544Z --> -

Fundamental Principle

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Mathematical induction is a method used to prove that a statement holds true for all natural numbers k.

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Let P(n) be a statement defined for each positive integer n.

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Then P(n) will be true for all positive integers n if the following two conditions are satisfied:

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(1) P(1) is true.

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(2) P(k) is true for some integer k + 1 implies that P(k + 1) is true.

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Because all the other values can use the result from the (1) and (2),

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Therefore, you can conclude that all the conditions with integers n are true.

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Fundamental Principle

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Mathematical induction is a method used to prove that a statement holds true for all natural numbers k.

+

Let P(n) be a statement defined for each positive integer n.

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Then P(n) will be true for all positive integers n if the following two conditions are satisfied:

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(1) P(1) is true.

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(2) P(k) is true for some integer k + 1 implies that P(k + 1) is true.

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Because all the other values can use the result from the (1) and (2),

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Therefore, you can conclude that all the conditions with integers n are true.

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E1 Example

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Q Prove the following for all positive n:

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A For n = 1, 

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The statement is true for n = 1.

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For n = k + 1, Assume the statement is true for some integer k >= 1.

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The statement is true for n = k + 1. By the principle of mathematical induction, the statement is true for all positive integers n.