# NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

NCERT solutions for exercise 4.1 Class 11 Maths chapter 4 deals with the principle of mathematical induction and related problems. Exercise 4.1 Class 11 Maths deals with how to prove a given mathematical statement using the principle of mathematical induction. NCERT solutions for Class 11 Maths chapter 4 exercise 4.1 gives an insight into the steps of proving a given statement using the idea of induction. Solving both example questions and Class 11 Maths chapter 4 exercise 4.1 are important to understand the concepts discussed in the chapter. Before starting the Class 11 Maths chapter 4 exercise 4.1 the NCERT book discusses the concept of deduction in brief and introduces what is induction and how induction is different from deduction. Followed by examples and Class 11 Maths chapter 4 exercise 4.1.

**Also see-**

**Principle Of Mathematical Induction Class 11 Chapter4**** ****Exercise 4.1**

**Exercise 4.1**

## Question:1 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:2 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:3 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:4 Prove the following by using the principle of mathematical induction for all :

Answer:

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:5 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:6 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by principle of mathematical induction , statement p(n) is true for all natural numbers n

Question:7 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:8 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have , which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:9 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:10 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:11 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:12 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:13 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:14 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

&nbsnbsp;

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:15 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:16 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:17 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:18 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:19 Prove the following by using the principle of mathematical induction for all : is a multiple of

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is multiple of 3, hence true

For n = k we have

, Let's assume that this is multiple of 3 = 3m

Now,

For n = k + 1 we have

Where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is multiple of 3 for all natural numbers n

Question:20 Prove the following by using the principle of mathematical induction for all : is a divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 11, hence true

For n = k we have

, Let's assume that this is divisible by 11 = 11m

Now,

For n = k + 1 we have

nbsp;

Where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 11 for all natural numbers n

Question:21 Prove the following by using the principle of mathematical induction for all : is divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by , hence true

For n = k we have

, Let's assume that this is divisible by

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by for all natural numbers n

Question:22 Prove the following by using the principle of mathematical induction for all : is divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 8, hence true

For n = k we have

, Let's assume that this is divisible by 8 = 8m

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 8 for all natural numbers n

Question:23 Prove the following by using the principle of mathematical induction for all : is a multiple of

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 27, hence true

For n = k we have

, Let's assume that this is divisible by 27 = 27m

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 27 for all natural numbers n

Question:24 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

**More About NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1**

Before the introduction of exercise 4.1 Class 11 Maths 8 example problems are given in NCERT chapter 4 of Class 11. Only one exercise is discussed in this chapter, that is Class 11 Maths chapter 4 exercise 4.1. 24 questions are discussed in the Class 11 Maths chapter 4 exercise 4.1. Solving all these questions are important as students may face similar or same questions in the exam paper. Give a good number of try to solve questions before looking to the NCERT solutions for Class 11 Maths chapter 4 exercise 4.1.

**Also Read| **Principle Of Mathematical Induction Class 11th Notes

**Benefits of NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1**

Solving all the questions of exercise 4.1 Class 11 Maths gives a good idea of the steps involved in solving a given problem using mathematical induction

From this chapter, students can definitely expect a question for the Class 11 final exam. To solve questions in the exam paper it is important to practice NCERT solutions for Class 11 Maths chapter 4 exercise 4.1.

**NCERT Solutions of Class 11 Subject Wise**

- NCERT Solutions for Class 11 Maths
- NCERT Solutions for Class 11 Physics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Biology

**Subject Wise NCERT Exampler Solutions**

- NCERT Exemplar Solutions for Class 11 Maths
- NCERT Exemplar Solutions for Class 11 Physics
- NCERT Exemplar Solutions for Class 11 Chemistry
- NCERT Exemplar Solutions for Class 11 Biology

## Frequently Asked Question (FAQs) - NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

**Question: **Why do we solve NCERT Solutions for Class 11 Maths chapter 4 exercise 4.1?

**Answer: **

The basic aim of solving the NCERT exercise is to understand the concepts well and to get an insight in to different types of questions that uses the concepts studied. Other than this Class 11th Maths chapter 4 exercise 4.1 will help in exams also.

**Question: **Which mathematician is credited with the origin of the principle of mathematical induction?

**Answer: **

Blaise Pascal

**Question: **Who was the first person to name and define mathematical induction?

**Answer: **

De Morgan

**Question: **What is the relation between the principle of mathematical induction and Peano’s axioms?

**Answer: **

It is a restatement of one of Peano’s axioms

**Question: **What number of solved examples are given in the NCERT Class 11 Maths Chapter 4 - Principle of Mathematical Induction?

**Answer: **

Eight examples are solved before the Exercise 4.1 Class 11 Maths

**Question: **How many questions are solved in the Class 11 Maths chapter 4 exercise 4.1?

**Answer: **

24 questions

**Question: **What are the three topics given in the chapter principle of mathematical induction?

**Answer: **

Introduction, motivation and the principle of mathematical induction

**Question: **How many topics are given in the NCERT Class 11 chapter 4?

**Answer: **

Three topics

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