## PRINCIPLE OF MATHEMATICAL INDUCTION

Statement: A sentence which is either exactly true or exactly false is called a statement.

Example: “26 is divisible by 5” is a statement.

“25 is divisible by 2 and 5” is a statement.

By Principle of Mathematical Induction, we can prove a mathematical statement true or false.

If a statement satisfies the following three steps, the statement will be correct.

Step 1: To prove a statement P(n) is true for first natural number1. [P(1) is true].

Step 2: To consider the statement P(n) is true for any other natural number. [Let P(m) is true].

Step 3: To prove the statement P(n) is true for a natural number (m + 1). [To prove P(m + 1) is true]

Example 1:

Prove that .

Solution:

Step 1:

Step 2: Let P(m) is true.

Step 3: To prove P(m+1) is true.

Example 2: Prove that: is divisible by 576.

Solution:

Step 1:

576 is divisible by 576.

Step 2: Let P(m) is true.

Step 3: To prove: P(m+1) is true.

is divisible by 576.

Hence P(n) is true for

Example 3:

Solution:

Step 1:

is true.

Step 2: Let P(m) is true.

Step 3: To prove P(m+1) is true.

From equation (1)

Multiplying both sides by

is true.

Here, P(n) is true for .