# Principle of Mathematical Induction Chapter 4 Class 11 Maths NCERT Textbook With Solutions PDF

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### Chapter 4: Principle of Mathematical Induction

4.1 Introduction

One key basis for mathematical thinking is deductive reasoning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed in three statements:

(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.

If statements (a) and (b) are true, then the truth of (c) is established. To make this simple mathematical example, we could write:

(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number, therefore,
(iii) Eight is an even number.

Thus, deduction, in a nutshell, is given a statement to be proven, often called a conjecture or a theorem in mathematics, valid deductive steps are derived and proof may or may not be established, i.e., the deduction is the application of a general case to a particular case.

In contrast to deduction, inductive reasoning depends on working with each case and developing a conjecture by observing incidences till we have observed each and every case.

It is frequently used in mathematics and is a key aspect of scientific reasoning, where collecting and analyzing data is the norm. Thus, in simple language, we can say the word induction means the generalization of particular cases or facts.

#### 4.3 The Principle of Mathematical Induction

Suppose there is a given statement P(n) involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then
the statement is also true for n = k + 1, i.e., the truth of P(k) implies the truth of P (k + 1).

### NCERT Solutions Class 11 Maths Chapter 4 Principle of Mathematical Induction

Prove the following by using the principle of mathematical induction for all n ∈ N:

1.

Solution:

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.

2.

Solution:

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.

3.

Solution:

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.

4.

Solution:

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.

5.

Solution:

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.