# NCERT solutions for Class 11 Mathematics chapter 4 - Principle of Mathematical Induction [Latest edition]

## Chapter 4: Principle of Mathematical Induction

Exercise 4.1
Exercise 4.1 [Pages 94 - 95]

### NCERT solutions for Class 11 Mathematics Chapter 4 Principle of Mathematical Induction Exercise 4.1 [Pages 94 - 95]

Exercise 4.1 | Q 1 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1 + 3 + 3^2 + ... + 3^(n – 1) =((3^n -1))/2

Exercise 4.1 | Q 2 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1^3 +  2^3 + 3^3 + ... + n^3 = ((n(n+1))/2)^2

Exercise 4.1 | Q 3 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1+ 1/((1+2)) + 1/((1+2+3)) +...+ 1/((1+2+3+...n)) = (2n)/(n +1)
Exercise 4.1 | Q 4 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2)  = (n(n+1)(n+2)(n+3))/(4(n+3))

Exercise 4.1 | Q 5 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.3 + 2.3^3 + 3.3^3  +...+ n.3^n = ((2n -1)3^(n+1) + 3)/4
Exercise 4.1 | Q 6 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.2 + 2.3 + 3.4+ ... + n(n+1) = [(n(n+1)(n+2))/3]

Exercise 4.1 | Q 7 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.3 + 3.5 + 5.7 + ...+(2n -1)(2n + 1) = (n(4n^2 + 6n -1))/3
Exercise 4.1 | Q 8 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

Exercise 4.1 | Q 9 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1/2 + 1/4 + 1/8 + ... + 1/2^n = 1 - 1/2^n

Exercise 4.1 | Q 10 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1/2.5 + 1/5.8 + 1/8.11 + ... + 1/((3n - 1)(3n + 2)) = n/(6n + 4)
Exercise 4.1 | Q 11 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/(n(n+1)(n+2)) = (n(n+3))/(4(n+1) (n+2))
Exercise 4.1 | Q 12 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

a + ar + ar^2 + ... + ar^(n -1) = (a(r^n - 1))/(r -1)
Exercise 4.1 | Q 13 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

(1+3/1)(1+ 5/4)(1+7/9)...(1 + ((2n + 1))/n^2) = (n + 1)^2

Exercise 4.1 | Q 14 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

(1+ 1/1)(1+ 1/2)(1+ 1/3)...(1+ 1/n) = (n + 1)

Exercise 4.1 | Q 15 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1^2 + 3^2 + 5^2 + ... + (2n -1)^2 = (n(2n - 1) (2n + 1))/3
Exercise 4.1 | Q 16 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1/1.4 + 1/4.7 + 1/7.10 + ... + 1/((3n - 2)(3n + 1)) = n/((3n + 1))

Exercise 4.1 | Q 17 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1/3.5 + 1/5.7 + 1/7.9 + ...+ 1/((2n + 1)(2n +3)) = n/(3(2n +3))
Exercise 4.1 | Q 18 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 1+2+ 3+...+n<1/8(2n +1)^2

Exercise 4.1 | Q 19 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ Nn (n + 1) (n + 5) is a multiple of 3.

Exercise 4.1 | Q 20 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11

Exercise 4.1 | Q 21 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ Nx2n – y2n is divisible by x y.

Exercise 4.1 | Q 22 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n– 9 is divisible by 8.

Exercise 4.1 | Q 23 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.

Exercise 4.1 | Q 24 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N (2+7) < (n + 3)2

Exercise 4.1

## NCERT solutions for Class 11 Mathematics chapter 4 - Principle of Mathematical Induction

NCERT solutions for Class 11 Mathematics chapter 4 (Principle of Mathematical Induction) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 4 Principle of Mathematical Induction are Motivation, Principle of Mathematical Induction.

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