NCERT Solutions | Class 11 Maths Chapter 4

NCERT Solutions | Class 11 Maths Chapter 4 | Principle of mathematical induction 

NCERT Solutions for Class 11 Maths Chapter 4 Principle of mathematical induction

CBSE Solutions | Maths Class 11

Check the below NCERT Solutions for Class 11 Maths Chapter 4 Principle of mathematical induction Pdf free download. NCERT Solutions Class 11 Maths  were prepared based on the latest exam pattern. We have Provided Principle of mathematical induction Class 11 Maths NCERT Solutions to help students understand the concept very well.

NCERT | Class 11 Maths

NCERT Solutions Class 11 Maths
Book: National Council of Educational Research and Training (NCERT)
Board: Central Board of Secondary Education (CBSE)
Class: 11th
Subject: Maths
Chapter: 4
Chapters Name: Principle of mathematical induction
Medium: English

Principle of mathematical induction | Class 11 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 11 Maths Chapter 4 Principle of mathematical induction to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

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

NCERT Exercises

Chapter 4 Principle of Mathematical Induction Exercise – 4.1

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 1NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 2

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

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 6

Ex 4.1 Class 11 Maths Question 4.
\(1.2.3+2.3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n\left( n+1 \right) \left( n+2 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }{ 4 } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(1.2.3+2.3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n\left( n+1 \right) \left( n+2 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }{ 4 } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 7 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 8

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 10

Ex 4.1 Class 11 Maths Question 6.
\(1.2+2.3+3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.\left( n+1 \right) =\left[ \frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \right] \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(1.2+2.3+3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.\left( n+1 \right) =\left[ \frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \right] \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 11 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 12

Ex 4.1 Class 11 Maths Question 7.
\(1.3+3.5+5.7+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\left( 2n-1 \right) \left( 2n+1 \right) =\frac { n\left( { 4n }^{ 2 }+6n-1 \right) }{ 3 } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(1.3+3.5+5.7+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\left( 2n-1 \right) \left( 2n+1 \right) =\frac { n\left( { 4n }^{ 2 }+6n-1 \right) }{ 3 } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 13 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 14

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 15

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 18

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 20

Ex 4.1 Class 11 Maths Question 11.
\(\frac { 1 }{ 1.2.3 } +\frac { 1 }{ 2.3.4 } +\frac { 1 }{ 3.4.5 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ n\left( n+1 \right) \left( n+2 \right) } =\frac { n\left( n+3 \right) }{ 4\left( n+1 \right) \left( n+2 \right) } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(\frac { 1 }{ 1.2.3 } +\frac { 1 }{ 2.3.4 } +\frac { 1 }{ 3.4.5 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ n\left( n+1 \right) \left( n+2 \right) } =\frac { n\left( n+3 \right) }{ 4\left( n+1 \right) \left( n+2 \right) } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 21 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 22 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 23

Ex 4.1 Class 11 Maths Question 12.
\(a+ar+{ ar }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ ar }^{ n-1 }=\frac { a\left( { r }^{ n }-1 \right) }{ r-1 } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(a+ar+{ ar }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ ar }^{ n-1 }=\frac { a\left( { r }^{ n }-1 \right) }{ r-1 } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 24 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 25

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 27

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 29

Ex 4.1 Class 11 Maths Question 15.
\({ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ \left( 2n-1 \right) }^{ 2 }=\frac { n\left( 2n-1 \right) \left( 2n+1 \right) }{ 3 } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \({ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ \left( 2n-1 \right) }^{ 2 }=\frac { n\left( 2n-1 \right) \left( 2n+1 \right) }{ 3 } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 30 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 31

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 32NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 33

Ex 4.1 Class 11 Maths Question 17.
\(\frac { 1 }{ 3.5 } +\frac { 1 }{ 5.7 } +\frac { 1 }{ 7.9 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 2n+1 \right) \left( 2n+3 \right) } =\frac { n }{ 3\left( 2n+3 \right) } \)
Solution.
Let the given statement be P(n), i.e.,
P(n) : \(\frac { 1 }{ 3.5 } +\frac { 1 }{ 5.7 } +\frac { 1 }{ 7.9 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 2n+1 \right) \left( 2n+3 \right) } =\frac { n }{ 3\left( 2n+3 \right) } \)
NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 34 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 35

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 37

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 NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction 39

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

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