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Mathematical Induction The Principle of Mathematical Induction Application in the Series Application in divisibility.

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Presentation on theme: "Mathematical Induction The Principle of Mathematical Induction Application in the Series Application in divisibility."— Presentation transcript:

1 Mathematical Induction The Principle of Mathematical Induction Application in the Series Application in divisibility

2 The Principle of Mathematical Induction 數學歸納法﹝ Mathematical Induction ﹞是用來証明某些與 自然數 n 有關的數學命題的一種方法。它的步驟是: 1. 驗証 n = 1 時命題成立﹝這叫歸納的基礎﹞; 2. 假設 n = k 時命題成立﹝這叫歸納假設﹞, 在這假設下証明 n= k+1 時命題成立。 根據 1 、 2 可以斷定命題對一切自然數都成立。

3 Application in Series Firstly, we need to know the names in the series clearly. The first term The n-th term

4 How many terms in the above series? Can you deduce the (n+1)-th term? Answer is:

5 Example Let the proposition is S(n),can you write down S(k)? Also, what is S(k+1)? k k WHO AM I ?

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7 In steps of calculation, the meaning of principle of mathematical induction is follows:

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9 Steps means:

10 Explanation Firstly, prove S(1) is true:

11 Assume S(k) is true, Use S(k) is true,prove S(k+1) is also true. Following steps is the hardest part in the mathematical induction.

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13 Because S(k) is true

14 By the principle of mathematical induction, S(n) is true for all positive integers n.

15 Example Let the proposition is S(n). When n = 1,

16 Assume S(k) is true, i.e., When n = k+1, By the principle of mathematical induction, S(n) is true for all positive integers n.

17 Application in divisibility The definition of divisibility If a and b be two integers with Then, Integer a is divisible by b if where M is an integer.

18 Example where 5 is an integer. where is an integer.

19 Explanation Prove, by M.I., is divisible by 8 for all natural numbers n. Let P(n) be the proposition P(1) is true

20 Assume that P(k) is true, M is an integer. When n = k+1,

21 9M + 1 is an integer. P(k+1) is true. By the principle of mathematical induction, is divisible by 8 for all natural numbers n.

22 Further Example Prove, by M.I., is divisible by 5 for all natural numbers n. Let P(n) be the proposition ‘ is divisible by 5. ’ Show P(1) is true.

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24 Assume that P(k) is true. where M is an integer. Consider P(k+1) is true or not. So hard!!

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26 k is positive integer. P(k+1) is true. By the principle of mathematical induction, is divisible by 5 for all natural numbers n. is an integer.


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