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Second Principle of Mathematical Induction

TopMathematical induction is a technique to prove a statement that if it is true or false for natural positive integers. Principle of Mathematical Induction is like game of dominoes; if one of the dominoes falls then the successive dominoes will also fall. It means that if one domino falls, second one will also fall followed by third one. There are basically three principles involved for proving mathematical induction.
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).