We have all studied mathematical induction, a very subtle mathematical method, which is mainly used to prove that a proposition is established within the scope of natural numbers. The general steps are as follows:

1: the proposition holds if n= 1;

2. Prove that if n=m is true, then it can be deduced that the proposition is also true when n=m+ 1.

3. It can be proved that this proposition is true.

This is our common mathematical induction. It is called the first induction. In fact, there is more than one form of mathematical induction, and there are many variants, but we can start with n=3, or just consider whether n is odd or even. There are the following complete induction methods:

1: Prove that the proposition p(n) holds when n = 1, 2, ..., k.

2. Prove that P (m), P (m+ 1), P (m+2), p(m+k- 1) holds, and we can deduce that p(m+k) holds, thus proving that this proposition holds. In other words, we will replace one in the first induction.

It is proved that when n= 1, 2, it can be tested.

Assume that when n=k and n=k+ 1, all propositions are true, that is:

This proves the correctness of this general formula. The content of mathematical induction is far more than what we learned in middle school. With this example, I hope students can open their eyes and explore the real kingdom of mathematics.