1. First Principle of Mathematical Induction is known as inductive base which proves that result is true for p = 1.

2. Second principle of mathematical induction is known as inductive hypothesis.

3. Third principle of mathematical induction is known as inductive step.

Let’s discuss about the second principle of mathematical induction:

Second principle of mathematical induction can be understood as shown below:

Assume that result is true for p = a, here we have to prove that result is true for p = a + 1.

Let’s take an example to understand second principle of mathematical induction:

Let us assume that series is 1 + 2 + 3 +………..+ p = p (p + 1)/2 for all positive integers ‘p’.

Step 1: We have to prove that result is true for p = 1.

L.H.S = p,

= (1),

= 1.

Now, R.H.S = p (p + 1)/2,

= 1(1 + 1)/2,

= 2/2,

= 1.

Step 2: Assume that result is true for p = a.

= 1 + 2 + 3 +………...+ a = a (a + 1)/2,

Now we have to prove that result is true for p = a + 1.

In this step we have to replace ‘p’ with (a + 1) on both the sides of the series. So,

We have to show 1 + 2 + 3 +………..+ (a + 1) = a + 1[(a + 1) + 1]/2,

L.H.S

1 + 2 + 3 +………..a + (a + 1),

From step 2;

a (a + 1)/2 + (a + 1),

(a + 1) (a/2 + 1).