Introduction to mathematical induction;
Mathematical induction is an important demonstration method. What we usually call "mathematical induction" mostly refers to its first form. Based on the principle of minimum (natural number), this paper makes a rough discussion on its second form, that is, the second mathematical induction, in order to deepen the understanding of mathematical induction and get an enhanced proof method.
Compared with the first mathematical induction, the second mathematical induction is more hypothetical, and what the first mathematical induction can prove in theory can certainly be proved by the second mathematical induction; On the contrary, not necessarily. We have a typical proof about the theory of integer division: "All integers greater than 1 can be decomposed into the product of several prime numbers." Look at this.
Mathematical induction description:
If the authenticity of the proposition when n=k+ 1 is proved, it must be based on the premise that n takes more than two or even all natural numbers not greater than k, then the second mathematical induction is generally used to prove it. The fundamental principle of this is that the inductive hypothesis of the second mathematical induction method is stronger than that of the first mathematical induction method.
It is not only required that the proposition holds when n=k, but also for all natural numbers less than k. Conversely, the mathematical proposition that can be proved by the first mathematical induction can certainly be proved by the second mathematical induction, which is not difficult to understand. But generally speaking, it is not necessary to do so.
The second mathematical induction, like the first mathematical induction, is also a manifestation of mathematical induction, which can prove that the second mathematical induction and the first mathematical induction are equivalent. The reason why we use different expressions is to facilitate our application.
Inductive reasoning:
Inductive reasoning is a kind of reasoning from individual to general. From a certain point of view about individual things to a larger point of view, the general principle and the explanation method of the principle are deduced from special concrete examples. What is in nature and society generally exists in the individual and the special, and exists through the individual.
Generally exists in specific objects and phenomena. Therefore, only by knowing the individual can we know the general. When explaining a larger thing, people sum up and generalize various universal principles or principles from individual and special things, and then they can draw conclusions about individual things from these principles and principles.