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
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:
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: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
, 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: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
, 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: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
, 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: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
, 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: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
, 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: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
, 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: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
, 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: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
, 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: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
, 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: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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