#### Chapters

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: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 4: Principle of Mathematical Induction

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 4 Principle of Mathematical Induction Solved Examples [Pages 61 - 70]

#### Short Answer Type

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

1 + 3 + 5 + ... + (2n – 1) = n^{2}^{ }

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`sum_(t = 1)^(n - 1) t(t + 1) = (n(n - 1)(n + 1))/3`, for all natural numbers n ≥ 2.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`(1 - 1/2^2).(1 - 1/3^2)...(1 - 1/n^2) = (n + 1)/(2n)`, for all natural numbers, n ≥ 2.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2^{2n} – 1 is divisible by 3.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2n + 1 < 2^{n}, for all natual numbers n ≥ 3.

#### Long Answer

Define the sequence a_{1}, a_{2}, a_{3} ... as follows:

a_{1 }= 2, a_{n} = 5 a_{n–1}, for all natural numbers n ≥ 2.

Write the first four terms of the sequence.

Define the sequence a_{1}, a_{2}, a_{3} ... as follows:

a_{1 }= 2, a_{n} = 5 a_{n–1}, for all natural numbers n ≥ 2.

Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula a_{n} = 2.5^{n–1} for all natural numbers.

The distributive law from algebra says that for all real numbers c, a_{1} and a_{2}, we have c(a_{1} + a_{2}) = ca_{1} + ca_{2}.

Use this law and mathematical induction to prove that, for all natural numbers, n ≥ 2, if c, a_{1}, a_{2}, ..., a_{n} are any real numbers, then c(a_{1} + a_{2} + ... + a_{n}) = ca_{1} + ca_{2} + ... + ca_{n}.

Prove by induction that for all natural number n sinα + sin(α + β) + sin(α + 2β)+ ... + sin(α + (n – 1)β) = `(sin (alpha + (n - 1)/2 beta)sin((nbeta)/2))/(sin(beta/2))`

Prove by the Principle of Mathematical Induction that 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

Show by the Principle of Mathematical Induction that the sum S_{n} of the n term of the series 1^{2} + 2 × 2^{2} + 3^{2} + 2 × 4^{2} + 5^{2} + 2 × 6^{2} ... is given by

S_{n} = `{{:((n(n + 1)^2)/2",", "if n is even"),((n^2(n + 1))/2",", "if n is odd"):}`

#### Objective Type Questions Choose the correct answer in Examples 11 and 12 (M.C.Q.)

Let P(n): “2^{n} < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer for which P(n) is true is ______.

1

2

3

4

A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.

For all n ∈ N

For all n > 5

For all n ≥ 5

For all n < 5

#### Fill in the blanks 13 and 14:

If P(n) : “2.4^{2n+1} + 3^{3n+1} is divisible by λ for all n ∈ N” is true, then the value of λ is ______.

If P(n): “49^{n} + 16^{n} + k is divisible by 64 for n ∈ N” is true, then the least negative integral value of k is ______.

State whether the following proof (by mathematical induction) is true or false for the statement.

P(n): 1^{2} + 2^{2} + ... + n^{2} = `(n(n + 1) (2n + 1))/6`

**Proof **By the Principle of Mathematical induction, P(n) is true for n = 1,

1^{2} = 1 = `(1(1 + 1)(2*1 + 1))/6`. Again for some k ≥ 1, k^{2} = `(k(k + 1)(2k + 1))/6`. Now we prove that

(k + 1)^{2} = `((k + 1)((k + 1) + 1)(2(k + 1) + 1))/6`

True

False

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 4 Principle of Mathematical Induction Exercise [Pages 70 - 72]

#### Short Answer

Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer

Give an example of a statement P(n) which is true for all n. Justify your answer.

**Prove the statement by using the Principle of Mathematical Induction:**

4^{n} – 1 is divisible by 3, for each natural number n.

**Prove the statement by using the Principle of Mathematical Induction:**

2^{3n} – 1 is divisible by 7, for all natural numbers n.

**Prove the statement by using the Principle of Mathematical Induction:**

n^{3} – 7n + 3 is divisible by 3, for all natural numbers n.

**Prove the statement by using the Principle of Mathematical Induction:**

3^{2n} – 1 is divisible by 8, for all natural numbers n.

**Prove the statement by using the Principle of Mathematical Induction:**

For any natural number n, 7^{n} – 2^{n} is divisible by 5.

**Prove the statement by using the Principle of Mathematical Induction:**

For any natural number n, x^{n} – y^{n} is divisible by x – y, where x and y are any integers with x ≠ y.

**Prove the statement by using the Principle of Mathematical Induction:**

n^{3} – n is divisible by 6, for each natural number n ≥ 2.

**Prove the statement by using the Principle of Mathematical Induction:**

n(n^{2} + 5) is divisible by 6, for each natural number n.

**Prove the statement by using the Principle of Mathematical Induction:**

n^{2} < 2^{n} for all natural numbers n ≥ 5.

**Prove the statement by using the Principle of Mathematical Induction:**

2n < (n + 2)! for all natural number n.

**Prove the statement by using the Principle of Mathematical Induction:**

`sqrt(n) < 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n)`, for all natural numbers n ≥ 2.

**Prove the statement by using the Principle of Mathematical Induction:**

2 + 4 + 6 + ... + 2n = n^{2} + n for all natural numbers n.

**Prove the statement by using the Principle of Mathematical Induction:**

1 + 2 + 2^{2} + ... + 2^{n} = 2^{n+1} – 1 for all natural numbers n.

**Prove the statement by using the Principle of Mathematical Induction:**

1 + 5 + 9 + ... + (4n – 3) = n(2n – 1) for all natural numbers n.

#### Long Answer Use the Principle of Mathematical Induction in the following

A sequence a_{1}, a_{2}, a_{3} ... is defined by letting a_{1} = 3 and a_{k} = 7a_{k – 1} for all natural numbers k ≥ 2. Show that a_{n} = 3.7^{n–1} for all natural numbers.

A sequence b_{0}, b_{1}, b_{2} ... is defined by letting b_{0} = 5 and b_{k} = 4 + b_{k – 1} for all natural numbers k. Show that b_{n} = 5 + 4n for all natural number n using mathematical induction.

A sequence d_{1}, d_{2}, d_{3} ... is defined by letting d_{1} = 2 and d_{k} = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that d_{n} = `2/(n!)` for all n ∈ N.

Prove that for all n ∈ N.

cos α + cos(α + β) + cos(α + 2β) + ... + cos(α + (n – 1)β) = `(cos(alpha + ((n - 1)/2)beta)sin((nbeta)/2))/(sin beta/2)`.

Prove that, cosθ cos2θ cos2^{2}θ ... cos2^{n–1}θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.

Prove that, sinθ + sin2θ + sin3θ + ... + sinnθ = `((sin ntheta)/2 sin ((n + 1))/2 theta)/(sin theta/2)`, for all n ∈ N.

Show that `n^5/5 + n^3/3 + (7n)/15` is a natural number for all n ∈ N.

Prove that `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24`, for all natural numbers n > 1.

Prove that number of subsets of a set containing n distinct elements is 2^{n}, for all n ∈ N.

#### Objective Type Questions from 26 to 30

If 10^{n} + 3.4^{n+2} + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.

5

3

7

1

For all n ∈ N, 3.5^{2n+1} + 2^{3n+1} is divisible by ______.

19

17

23

25

If x^{n} – 1 is divisible by x – k, then the least positive integral value of k is ______.

1

2

3

4

#### Fill in the blanks in the following:

If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.

**State whether the following statement is true or false. Justify.**

Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.

True

False

## Chapter 4: Principle of Mathematical Induction

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

NCERT solutions for Mathematics Exemplar Class 11 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 Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

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