What is Mathematical Induction?

Mathematical Induction is one of the fundamental methods of writing proofs and it is used to prove a given statement about any well-organized set. Generally, it is used for proving results or establishing statements that are formulated in terms of n, where n is a natural number.

Suppose P(n) is a statement for n natural number then it can be proved using the Principle of Mathematical Induction, Firstly we will prove for P(1) then let P(k) be true then prove for P(k+1). If P(k+1) holds then we say that P(n) is true by the principle of mathematical induction.

We can compare mathematical induction to falling dominoes. When a domino falls, it knocks down the next domino in succession. The first domino knocks down the second one, the second one knocks down the third, and so on. In the end, all of the dominoes will be bowled over. But there are some conditions to be fulfilled:

• Base step is that the starting domino must fall to set the knocking process in action.
• The distance between dominoes must be equal for any two adjacent dominoes. Otherwise, a certain domino may fall down without bowling over the next. Then the sequence of reactions will stop. Maintaining the equal inter-domino distance ensures that P(k) ⇒ P(k + 1) for each integer k ≥ a. This is the inductive step.

Mathematical induction is a concept in mathematics that is used to prove various mathematical statements and theorems. The principle of mathematical induction is sometimes referred to as PMI. It is a technique that is used to prove the basic theorems in mathematics which involve the solution up to n finite natural terms. 

Principle of Mathematical Induction is widely used in proving various statements such as a sum of first n natural numbers is given by the formula n(n+1)/2. This can be easily proved using the Principle of Mathematical Induction.

In this article, we will learn about the principle of mathematical induction, its statement, its example, and others in detail.