Abstract
All mathematical research is based on deductive and inductive methods. The deductive method of reasoning is a reasoning from the general to the particular, i.e., a reasoning whose starting point is the general result, and the final point is the particular one. Induction is used in the transition from particular results to general ones, i.e. it is the opposite of the deductive method. The paper considers the possibilities of applying the necessary and effective method of proving mathematical statements – the method of mathematical induction to solving problems of various branches of mathematics. This article describes the applications of the mathematical induction method for proving various identities, inequalities, for calculating the nth degree of matrices of different orders, derivatives of n-th order functions, determinants of n-th order, some improper integrals of the 1st kind, for proving statements on divisibility.