# Principle of Mathematical Induction  ## Principle of Mathematical Induction:

Often it becomes a difficult job to prove a mathematical statement by a direct method. In such a case, an indirect method, known as the method of mathematical induction is used. This is a special technique to prove a mathematical statement involving a variable n, usually denoting a positive integer, in three steps. The steps are-

Verification Step- At this stage, the mathematical statement is verified to be true for the smallest permissible value of n.

Assumption Step- Here, the statement is assumed to be true for n = m, where m is any positive integer greater than the permissible lowest value of n.

Induction Step- This may also be referred to as the concluding step where the statement, which has been assumed to be true for n = m, is shown to be true for n = m + 1 only.

Now, as the statement has already been verified for n = 1 (or for the lowest permissible value of n), following the induction step, it will be true for n = 2 and so on. Thus, logically the statement is proved to be true for any value of n.

Hence, the principle of mathematical induction states that “If P(n) be a statement such that P(1) is verified to be true and P(m + 1) is true whenever P(m) is true, m being a positive integer, then the statement is true for all positive integral values of n.