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

-

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.

-

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

-

(1) P(1) is true.

-

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

-

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

-

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

+

Fundamental Principle

+

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.

+

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

+

(1) P(1) is true.

+

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

+

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

+

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

+

 

+

E1 Example

+

Q Prove the following for all positive n:

+
+

A For n = 1, 

+
+

The statement is true for n = 1.

+

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

+
+
+
+
+

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