NCERT Class 11-Math՚S: Chapter – 4 Principle of Mathematical Induction Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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4.1 Overview

Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of , where is a positive integer.

4.1. 1 The principle of mathematical induction

Let be a given statement involving the natural number n such that

(i) The statement is true for is true (or true for any fixed natural number) and

(ii) If the statement is true for (where is a particular but arbitrary natural number) , then the statement is also true for , i.e., truth of P (k) implies the truth of . Then is true for all natural numbers .

4.2 Solved Examples

Short Answer Type

Prove statements in Examples 1 to 5, by using the Principle of Mathematical Induction for all , that:

Question 1:

Answer:

Let the given statement be defined as , for . Note that is true, since

Assume that is true for some

Now, to prove that is true, we have

Thus, is true, whenever is true.

Hence, by the Principle of Mathematical Induction, is true for all .

Question 2:

Answer:

We observe that

Thus, in true for .

Assume that is true for .

To prove that is true, we have

Thus, is true, whenever is true.

Hence, by the Principle of Mathematical Induction, is true for all natural numbers .

Question 3:

for all natural numbers, .

Answer:

Let the given statement be , i.e.. ,

for all natural numbers, We, observe that is true, since

Assume that is true for some , i.e.. ,

Now, to prove that is true, we have

Thus, is true, whenever is true.

Hence, by the Principle of Mathematical Induction, is true for all natural numbers, .