Method of mathematical induction

The method of mathematical induction can be equated to progress. So, starting from the lowest level, researchers with the help of logical thinking pass to the higher. Any self-respecting person constantly strives for progress and the ability to think logically. That is why inductive thinking was created by nature.

The term "induction" in translation into Russian means guidance, therefore it is considered inductive that conclusions are based on the results of certain experiments and observations, which are obtained by forming from the particular to the general.

An example is the contemplation of the sunrise. After observing 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 are widely used and applied in experimental sciences. Thus, with the help of them, we can formulate propositions on the basis of which further deductions can be made by means of deductive methods . With certain certainty it can be argued that the "three whales" of theoretical mechanics-the laws of Newton's motion-are themselves the result of carrying out private experiments with summing up the total. Kepler's law on the motion of the planets was derived by him on the basis of many years of observations of T. Braga, a Danish astronomer. It is in these cases that induction played a positive role in refining and generalizing the assumptions made.

Despite the expansion of the field of its application, the method of mathematical induction, unfortunately, takes little time in the school curriculum. However, in the modern world it is from childhood that it is necessary to train the younger generation to think inductively, and not simply to solve problems according to a certain pattern or a 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 set of numbers depending on the natural variables.

The principle of mathematical induction is based on proving the truth of the sentence A (n) for any values of a variable and consists of two stages:

1. The truth of the proposition A (n) is proved for n = 1.

2. In the case where the sentence A (n) remains true for n = k (k is a natural number), it will be true for the next value n = k + 1.

This principle also formulates the method of mat. Induction. Often it is accepted as an axiom that defines a number of numbers, and is applied without evidence.

There are times when the method of mathematical induction is in some cases 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 the truthfulness of A (1);

- to prove the truth of the statement A (k + 1) when taking into account the truth of A (k).

In the case of a successful proof of the validity of this proposition, A (n) for all values of n is considered true for any positive integer k, in accordance with this principle.

The above method of mathematical induction is widely used in the proofs of identities, theorems, inequalities. It can also be used in solving geometrical problems 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 derived from axioms. However, it is possible to formulate a large number of statements from these axioms. And it is the choice of statements that is prompted by the use of induction. With the help of this method it is possible to divide all the theorems into necessary for science and practice and not very much.