The method of mathematical induction can be equated with progress. So, starting from the lowest level, researchers using logical thinking move to the higher. Any self-respecting person is constantly striving for progress and the ability to think logically. That is why nature created inductive thinking.
The term “induction” in translation into Russian means guidance, therefore, it is customary to consider conclusions based on the results of certain experiments and observations that are obtained by forming from particular to general as inductive.
An example is the contemplation of a sunrise. Having observed this phenomenon for several days in a row, we can say that from the east the sun will rise tomorrow, and the day after tomorrow, etc.
Inductive conclusions have been widely used and are used in experimental sciences. So, with the help of them it is possible to formulate the provisions on the basis of which further conclusions can already be made using deductive methods . With some certainty, it can be argued that the “three pillars” of theoretical mechanics — the laws of Newton’s motion — are themselves the result of conducting private experiments with a summary. And Kepler’s law on planetary motion was derived by him on the basis of many years of observations by T. Brahe, a Danish astronomer. It was in these cases that induction played a positive role in clarifying and generalizing the assumptions made.
Despite the expansion of its field of application, the method of mathematical induction, unfortunately, takes little time in the school curriculum. However, in the modern world, it is precisely from childhood that it is necessary to accustom the younger generation to think inductively, and not just solve problems according to a certain pattern or given formula.
The method of mathematical induction can be widely applied in algebra, arithmetic, and geometry. In these sections, it is necessary to prove the truth of a certain set of numbers, depending on natural variables.
The principle of mathematical induction is based on the proof of the truth of the sentence A (n) for any values of the variable and consists of two stages:
1. The truth of the sentence A (n) is proved for n = 1.
2. In the case when the sentence A (n) remains true for n = k (k is a positive integer), it will be true for the next value n = k + 1.
This principle and formulates the method of mat. induction. Often it is accepted as an axiom that defines a series of numbers, and is applied without evidence.
There are times when the method of mathematical induction in some cases is subject to proof. So, in the case when it is required to prove the truth of the proposed set A (n) for all natural numbers n, it is necessary:
- check for truth the statement A (1);
- prove the truth of the statement A (k + 1) while taking into account the truth of A (k).
In the case of a successful proof of the validity of this sentence for any natural number k, the sentence A (n) is recognized true for all values of n, in accordance with the indicated principle.
The above method of mathematical induction is widely used in the proofs of identities, theorems, inequalities. It can also be used in solving problems of a geometric nature and divisibility.
However, one should not think that this ends the use of the induction method in mathematics. For example, it is not necessary to experimentally verify all theorems that are logically inferred from axioms. But at the same time, of these axioms there is the possibility of formulating a large number of statements. And it is precisely the choice of statements that is suggested by the use of induction. Using this method, you can divide all the theorems into those necessary for science and practice, and not so much.