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

 

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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. Another introductory concept of NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction is Inductive reasoning. 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. 

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

The main topic of  NCERT Solutions of this chapter is the Principle of Mathematical Induction, which is 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.

The above steps are followed to solve the problems in NCERT Class 11 Maths Chapter 4. For example:

Q: 1) Prove the following by using the principle of mathematical induction for all  n\in N:

       1+3+3^2+...+3^{n-1}=\frac{(3^n-1)}{2}

Solution: Let the given statement be p(n) i.e.
p(n):1+3+3^2+...+3^{n-1}=\frac{(3^n-1)}{2}
For n = 1  we have
p(1): 1=\frac{(3^1-1)}{2}=\frac{3-1}{2}= \frac{2}{2}=1    ,   which is true

For  n = k  we have
p(k):1+3+3^2+...+3^{k-1}=\frac{(3^k-1)}{2} \ \ \ \ \ \ \ -(i)   ,        Let's assume that this statement is true

Now,
For  n = k + 1  we have
p(k+1):1+3+3^2+...+3^{k+1-1}= 1+3+3^2+...+3^{k-1}+3^{k}
                                                                           = (1+3+3^2+...+3^{k-1})+3^{k}
                                                                           = \frac{(3^k-1)}{2}+3^{k} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Using \ (i))
                                                                           = \frac{3^k-1+2.3^k}{2}
                                                                           = \frac{3^k(1+2)-1}{2}
                                                                           = \frac{3.3^k-1}{2}
                                                                           = \frac{3^{k+1}-1}{2}

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

There are 24 problems mentioned in the NCERT Solutions for Class 11. Solving all the questions of NCERT exercise are mandatory and below are the NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction.

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

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

NCERT Solutions for Class 11 Maths- Chapter-Wise

Chapter-1

Sets

Chapter-2

Relations and Functions

Chapter-3

Trigonometric Functions

Chapter-4

Principle of Mathematical Induction

Chapter-5

Complex Numbers and Quadratic equations

Chapter-6

Linear Inequalities

Chapter-7

Permutation and Combinations

Chapter-8

Binomial Theorem

Chapter-9

Sequences and Series

Chapter-10

Straight Lines

Chapter-11

Conic Section

Chapter-12

Introduction to Three Dimensional Geometry

Chapter-13

Limits and Derivatives

Chapter-14

Mathematical Reasoning

Chapter-15

Statistics

Chapter-16

Probability

NCERT Solutions for Class 11- Subject Wise

NCERT Solutions for Class 11 Biology

NCERT Solutions for Class 11 Maths

NCERT Solutions for Class 11 Chemistry

NCERT Solutions for Class 11 Physics

 

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