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

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction: In this chapter, you will learn a mathematical technique that can be used to prove a theorem, formula or statement true for every natural number. It is called mathematical induction. In this article, you will get NCERT solutions for class 11 maths chapter 4 principle of mathematical induction. This chapter starts with an introduction to deductive and inductive reasoning.

Deductive reasoning uses some statement of facts and deducts another fact.

For example, consider the following statements a and b are true

a) All men are mortal

b) Raju is a man

From the above two statements we can deduct the fact c) Raju is mortal that is using a and b, c is established.

In inductive reasoning, we look for a pattern and obtaining a conclusion. That is induction means generalization from particular cases. For example, if we have data on the population of a city for the last 10 years. Based on the trend of population growth in the last 10 years we can predict the population of the next coming 2 years. This problem comes under induction. In CBSE NCERT solutions for class 11 maths chapter 4 principle of mathematical induction, you will learn a two-step process to prove any formula, statement or theorem. There are 24 problems in this chapter. All these problems are explained in solutions of NCERT for class 11 maths chapter 4 principle of mathematical induction in a step-by-step manner. It will be very easy for you to understand the concepts. Check all NCERT solutions from class 6 to 12 to learn CBSE science and maths.

## The main topics of NCERT Class 11 Maths Chapter 4 Principle of Mathematical Induction are

4.1 Introduction

4.2 Motivation

4.3 The Principle of Mathematical Induction

## NCERT solutions for class 11 maths chapter 4 principle of mathematical induction-Exercise: 4.1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## NCERT solutions for class 11 mathematics

 chapter-1 NCERT solutions for class 11 maths chapter 1 Sets chapter-2 Solutions of NCERT for class 11 chapter 2 Relations and Functions chapter-3 CBSE NCERT solutions for class 11 chapter 3 Trigonometric Functions chapter-4 NCERT solutions for class 11 Maths chapter 4 Principle of Mathematical Induction chapter-5 Solutions of NCERT for class 11 chapter 5 Complex Numbers and Quadratic equations chapter-6 CBSE NCERT solutions for class 11 maths chapter 6 Linear Inequalities chapter-7 NCERT solutions for class 11 maths chapter 7 Permutation and Combinations chapter-8 Solutions of NCERT for class 11 maths chapter 8 Binomial Theorem chapter-9 CBSE NCERT solutions for class 11 maths chapter 9 Sequences and Series chapter-10 NCERT solutions for class 11 maths chapter 10 Straight Lines chapter-11 Solutions of NCERT for class 11 maths chapter 11 Conic Section chapter-12 CBSE NCERT solutions for class 11 maths chapter 12 Introduction to Three Dimensional Geometry chapter-13 NCERT solutions for class 11 maths chapter 13 Limits and Derivatives chapter-14 Solutions of NCERT for class 11 maths chapter 14 Mathematical Reasoning chapter-15 CBSE NCERT solutions for class 11 maths chapter 15 Statistics chapter-16 NCERT solutions for class 11 maths chapter 16 Probability

## NCERT solutions for class 11- Subject wise

 Solutions of NCERT for class 11 biology CBSE NCERT solutions for class 11 maths NCERT solutions for class 11 chemistry Solutions of NCERT for Class 11 physics

All the above questions explained in NCERT solutions for class 11 maths chapter 4 principle of mathematical induction followed the same common process stated as given below.

Suppose there is a  statement q(n) involving the natural number n

such that (i) The statement is true for n = 1, that is, q(1) is true

(ii) If the statement is true for n = k (k is some positive integer), then the statement is also true for n = k + 1

Note:  Property (ii)  does not assert that the given statement is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1.

There are 24 problems solved in this article. First, try to solve by yourself. If you are finding difficulties, you can take help from CBSE NCERT solutions for class 11 maths chapter 4 principle of mathematical induction.

Happy learning !!!