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

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。

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

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

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

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

Steps means:

Explanation Firstly, prove S(1) is true:

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.

Because S(k) is true

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

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

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.

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.

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

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

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

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.

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.

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

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.