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Class 11th Chapter -4 Principle of Mathematical Induction | NCERT Maths Solution | NCERT Solution | Edugrown

In This Post we are providing Chapter – 4 Principle of Mathemarical Induction NCERT Solutions for Class 11 Maths which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These Class 11 Principle of Mathemarical Induction solutions can be really helpful in the preparation of Board exams and will provide you with in depth detail of the chapter.

We have solved every question stepwise so you don’t have to face difficulty in understanding the solutions. It will also give you concepts that are important for overall development of students. Class 11 Maths Principle of Mathemarical Induction NCERT Written Solutions will be useful in higher classes as well because variety of questions related to these concepts can be asked so you must study and understand them properly.

Class 11th Chapter 4 Principle of Mathemarical Induction| NCERT MATHS SOLUTION |

Chapter 4 Principle of Mathemarical Induction EX 4.1 NCERT Solutions

Prove the following by using the principle of mathematical induction for aline n ∈ N :
Ex 4.1 Class 11 Maths Question 1.
$1+{ 3 }^{ 2 }+{ 3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ 3 }^{ n }=frac { left( { 3 }^{ n }-1 right) }{ 2 }$

Solution.
Let the given statement be P(n) i.e.,
P(n) : $1+{ 3 }^{ 2 }+{ 3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ 3 }^{ n }=frac { left( { 3 }^{ n }-1 right) }{ 2 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 1

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 1

Ex 4.1 Class 11 Maths Question 2.
${ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ n }^{ 3 }={ left( frac { nleft( n+1 right) }{ 2 } right) }^{ 2 }$
Solution.
Let the given statement be P(n) i.e.,
P(n) : ${ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ n }^{ 3 }={ left( frac { nleft( n+1 right) }{ 2 } right) }^{ 2 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 3

Ex 4.1 Class 11 Maths Question 3.
$1+frac { 1 }{ left( 1+2 right) } +frac { 1 }{ left( 1+2+3 right) } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 1+2+3+cdot cdot cdot cdot cdot cdot cdot cdot +n right) } =frac { 2 }{ left( n+1 right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1+frac { 1 }{ left( 1+2 right) } +frac { 1 }{ left( 1+2+3 right) } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 1+2+3+.cdot cdot cdot cdot cdot cdot cdot cdot +n right) } =frac { 2 }{ left( n+1 right) }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 5

Ex 4.1 Class 11 Maths Question 4.
$1.2.3+2.3.4+cdot cdot cdot cdot cdot cdot cdot cdot +nleft( n+1 right) left( n+2 right) =frac { nleft( n+1 right) left( n+2 right) left( n+3 right) }{ 4 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2.3+2.3.4+cdot cdot cdot cdot cdot cdot cdot cdot +nleft( n+1 right) left( n+2 right) =frac { nleft( n+1 right) left( n+2 right) left( n+3 right) }{ 4 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 7

Ex 4.1 Class 11 Maths Question 5.
$1.3+{ 2.3 }^{ 2 }+{ 3.3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ n.3 }^{ n }=frac { left( 2n-1 right) { 3 }^{ n+1 }+3 }{ 4 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.3+{ 2.3 }^{ 2 }+{ 3.3 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ n.3 }^{ n }=frac { left( 2n-1 right) { 3 }^{ n+1 }+3 }{ 4 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 9

Ex 4.1 Class 11 Maths Question 6.
$1.2+2.3+3.4+cdot cdot cdot cdot cdot cdot cdot cdot +n.left( n+1 right) =left[ frac { nleft( n+1 right) left( n+2 right) }{ 3 } right]$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2+2.3+3.4+cdot cdot cdot cdot cdot cdot cdot cdot +n.left( n+1 right) =left[ frac { nleft( n+1 right) left( n+2 right) }{ 3 } right]$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 11

Ex 4.1 Class 11 Maths Question 7.
$1.3+3.5+5.7+cdot cdot cdot cdot cdot cdot cdot cdot +left( 2n-1 right) left( 2n+1 right) =frac { nleft( { 4n }^{ 2 }+6n-1 right) }{ 3 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.3+3.5+5.7+cdot cdot cdot cdot cdot cdot cdot cdot +left( 2n-1 right) left( 2n+1 right) =frac { nleft( { 4n }^{ 2 }+6n-1 right) }{ 3 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 13

Ex 4.1 Class 11 Maths Question 8.
$1.2+2.{ 2 }^{ 2 }+3.{ 2 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +n.{ 2 }^{ n }=left( n-1 right) { 2 }^{ n+1 }+2$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2+2.{ 2 }^{ 2 }+3.{ 2 }^{ 3 }+cdot cdot cdot cdot cdot cdot cdot cdot +n.{ 2 }^{ n }=left( n-1 right) { 2 }^{ n+1 }+2$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 16

Ex 4.1 Class 11 Maths Question 9
$frac { 1 }{ 2 } +frac { 1 }{ 4 } +frac { 1 }{ 8 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ { 2 }^{ n } } =1-frac { 1 }{ { 2 }^{ n } }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $frac { 1 }{ 2 } +frac { 1 }{ 4 } +frac { 1 }{ 8 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ { 2 }^{ n } } =1-frac { 1 }{ { 2 }^{ n } }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 17

Ex 4.1 Class 11 Maths Question 10.
$frac { 1 }{ 2.5 } +frac { 1 }{ 5.8 } +frac { 1 }{ 8.11 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 3n-1 right) left( 3n+2 right) } =frac { n }{ left( 6n+4 right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $frac { 1 }{ 2.5 } +frac { 1 }{ 5.8 } +frac { 1 }{ 8.11 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 3n-1 right) left( 3n+2 right) } =frac { n }{ left( 6n+4 right) }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 19

Ex 4.1 Class 11 Maths Question 11.
$frac { 1 }{ 1.2.3 } +frac { 1 }{ 2.3.4 } +frac { 1 }{ 3.4.5 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ nleft( n+1 right) left( n+2 right) } =frac { nleft( n+3 right) }{ 4left( n+1 right) left( n+2 right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $frac { 1 }{ 1.2.3 } +frac { 1 }{ 2.3.4 } +frac { 1 }{ 3.4.5 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ nleft( n+1 right) left( n+2 right) } =frac { nleft( n+3 right) }{ 4left( n+1 right) left( n+2 right) }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 21

Ex 4.1 Class 11 Maths Question 12.
$a+ar+{ ar }^{ 2 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ ar }^{ n-1 }=frac { aleft( { r }^{ n }-1 right) }{ r-1 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $a+ar+{ ar }^{ 2 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ ar }^{ n-1 }=frac { aleft( { r }^{ n }-1 right) }{ r-1 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 24

Ex 4.1 Class 11 Maths Question 13.
$left( 1+frac { 3 }{ 1 } right) left( 1+frac { 5 }{ 4 } right) left( 1+frac { 7 }{ 9 } right) cdot cdot cdot cdot cdot cdot cdot cdot left( 1+frac { left( 2n+1 right) }{ { n }^{ 2 } } right) ={ left( n+1 right) }^{ 2 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $left( 1+frac { 3 }{ 1 } right) left( 1+frac { 5 }{ 4 } right) left( 1+frac { 7 }{ 9 } right) cdot cdot cdot cdot cdot cdot cdot cdot left( 1+frac { left( 2n+1 right) }{ { n }^{ 2 } } right) ={ left( n+1 right) }^{ 2 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 26

Ex 4.1 Class 11 Maths Question 14.
$left( 1+frac { 1 }{ 1 } right) left( 1+frac { 1 }{ 2 } right) left( 1+frac { 1 }{ 3 } right) cdot cdot cdot cdot cdot cdot cdot cdot left( 1+frac { 1 }{ n } right) =left( n+1 right)$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $left( 1+frac { 1 }{ 1 } right) left( 1+frac { 1 }{ 2 } right) left( 1+frac { 1 }{ 3 } right) cdot cdot cdot cdot cdot cdot cdot cdot left( 1+frac { 1 }{ n } right) =left( n+1 right)$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 28

Ex 4.1 Class 11 Maths Question 15.
${ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ left( 2n-1 right) }^{ 2 }=frac { nleft( 2n-1 right) left( 2n+1 right) }{ 3 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : ${ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+cdot cdot cdot cdot cdot cdot cdot cdot +{ left( 2n-1 right) }^{ 2 }=frac { nleft( 2n-1 right) left( 2n+1 right) }{ 3 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 30

Ex 4.1 Class 11 Maths Question 16.
$frac { 1 }{ 1.4 } +frac { 1 }{ 4.7 } +frac { 1 }{ 7.10 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 3n-2 right) left( 3n+1 right) } =frac { n }{ left( 3n+1 right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $frac { 1 }{ 1.4 } +frac { 1 }{ 4.7 } +frac { 1 }{ 7.10 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 3n-2 right) left( 3n+1 right) } =frac { n }{ left( 3n+1 right) }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 32

Ex 4.1 Class 11 Maths Question 17.
$frac { 1 }{ 3.5 } +frac { 1 }{ 5.7 } +frac { 1 }{ 7.9 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 2n+1 right) left( 2n+3 right) } =frac { n }{ 3left( 2n+3 right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $frac { 1 }{ 3.5 } +frac { 1 }{ 5.7 } +frac { 1 }{ 7.9 } +cdot cdot cdot cdot cdot cdot cdot cdot +frac { 1 }{ left( 2n+1 right) left( 2n+3 right) } =frac { n }{ 3left( 2n+3 right) }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 34

Ex 4.1 Class 11 Maths Question 18.
$1+2+3+cdot cdot cdot cdot cdot cdot cdot cdot +n<frac { 1 }{ 8 } { left( 2n+1 right) }^{ 2 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1+2+3+cdot cdot cdot cdot cdot cdot cdot cdot +n<frac { 1 }{ 8 } { left( 2n+1 right) }^{ 2 }$
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 36

Ex 4.1 Class 11 Maths Question 19.
n(n+1 )(n + 5) is a multiple of 3.
Solution.
Let the given statement be P(n), i.e.,
P(n): n(n + l)(n + 5) is a multiple of 3.
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 38

Ex 4.1 Class 11 Maths Question 20.
${ 10 }^{ 2n-1 }+1$ is divisible by 11.
Solution.
Let the given statement be P(n), i.e.,
P(n): ${ 10 }^{ 2n-1 }+1$ is divisible by 11
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 40

Ex 4.1 Class 11 Maths Question 21.
${ x }^{ 2n }-{ y }^{ 2n }$ is divisible by x + y.
Solution.
Let the given statement be P(n), i.e.,
P(n): ${ x }^{ 2n }-{ y }^{ 2n }$ is divisible by x + y.
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 41

Ex 4.1 Class 11 Maths Question 22.
${ 3 }^{ 2n+2 }-8n-9$ is divisible by 8.
Solution.
Let the given statement be P(n), i.e.,
P(n): ${ 3 }^{ 2n+2 }-8n-9$ is divisible by 8.
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 42

Ex 4.1 Class 11 Maths Question 23.
${ 41 }^{ n }-{ 14 }^{ n }$ is a multiple of 27.
Solution.
Let the given statement be P(n), i.e.,
P(n): ${ 41 }^{ n }-{ 14 }^{ n }$ is a multiple of 27.
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 43

Ex 4.1 Class 11 Maths Question 24.
$left( 2n+7 right) <{ left( n+3 right) }^{ 2 }$
Solution.
Let the given statement be P(n), i.e.,
P(n): $left( 2n+7 right) <{ left( n+3 right) }^{ 2 }$
First we prove that the statement is true for n = 1.
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 44

We hope the NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction help you. If you have any query regarding NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction, drop a comment below and we will get back to you at the earliest.

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Class 11th Chapter -4 Principle of Mathematical Induction | NCERT Maths Solution | NCERT Solution | Edugrown

Class 11th Chapter 4 Principle of Mathemarical Induction| NCERT MATHS SOLUTION |

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