Mathematical induction is a method to prove related propositions of natural numbers, which is widely used. It is especially obvious in the college entrance examination papers in recent years. Here, we talk about the application of mathematical induction through several college entrance examination questions.

First, prove the divisibility problem by mathematical induction.

When proving the divisibility problem by mathematical induction, it is a great skill to prove the problem by mathematical induction. First, we pieced together a hypothetical formula from the formula to be proved, and then proved that the remaining formula can also be divisible by a certain formula (number).

Example 1. Is there a positive integer m that makes f(n)=(2n+7)? 3n+9 is divisible by m for any natural number n? If it exists, find the maximum m value to prove your conclusion; If it does not exist, please explain why.

Proof: Solution: From f(n)=(2n+7)? 3n+9, f( 1)=36, f(2)=3×36, f(3)= 10×36, f(4)=34×36, from which we can guess that m=36.

The following is proved by mathematical induction:

(1) When n= 1, it is obviously true.

(2) If n=k, f(k) can be divisible by 36, that is, f(k)=(2k+7)? 3k+9 is divisible by 36; When n=k+ 1, [2 (k+ 1)+7]? 3k+ 1+9=3[(2k+7)？ 3k+9]+ 18(3k- 1- 1)，

Since 3k- 1- 1 is a multiple of 2, 18 (3k- 1- 1) can be divisible by 36. That is to say, when n=k+ 1, f(n) can also be divisible by 36.

According to (1)(2), for all positive integers n, there is f(n)=(2n+7)? 3n+9 is divisible by 36, and the maximum value of m is 36.

Second, prove the identity problem by mathematical induction.

For the problem of proving identities, we should compare the conclusion with the derivation process in time when the equation is also established, which is what we usually call the method of combining two sides. This can reduce the complexity of calculation, find the formula to be proved, and make the proof of the problem purposeful.

Example 2: Is there a constant that makes the equation hold true for all natural numbers? And prove your conclusion.

Solution: Assuming that it exists, the equations in the problem are established, and they were also established at that time, so substitute.

Solution, so correct, the following equation holds:

manufacture

Assuming that the above formula holds, that is

therefore

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