Free download NCERT Solutions for Class 11 Maths Chapter 4** Principles of Mathematical Induction Ex 4.1 and Miscellaneous Exercise** PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Topics and Sub Topics in Class 11 Maths Chapter 4 Principle of Mathematical Induction:

Section Name | Topic Name |

4 | Principle of Mathematical Induction |

4.1 | Introduction |

4.2 | Motivation |

4.3 | The Principle of Mathematical Induction |

**Contents**show

### NCERT Solutions for Class 11 Maths Chapter 4

**Question 1:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): 1 + 3 + 3^{2} + …+ 3^{n}^{–1} =

For *n* = 1, we have

P(1): 1 =, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1 + 3 + 3^{2} + … + 3^{k}^{–1} + 3^{(}^{k}^{+1) – 1}

= (1 + 3 + 3^{2} +… + 3^{k}^{–1}) + 3^{k}

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 i.e., *n*.

**Question 2:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): 1^{3} = 1 =, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1^{3} + 2^{3} + 3^{3} + … + *k*^{3} + (*k* + 1)^{3}

= (1^{3} + 2^{3} + 3^{3} + …. + *k*^{3}) + (*k* + 1)^{3}

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 i.e., *n*.

**Question 3:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): 1 = which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

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 i.e., *n*.

**Question 4:**

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) =

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): 1.2.3 + 2.3.4 + … + *n*(*n* + 1) (*n* + 2) =

For *n* = 1, we have

P(1): 1.2.3 = 6 =, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

1.2.3 + 2.3.4 + … + *k*(*k* + 1) (*k* + 2)

We shall now prove that P(*k* + 1) is true.

Consider

1.2.3 + 2.3.4 + … + *k*(*k* + 1) (*k* + 2) + (*k* + 1) (*k* + 2) (*k* + 3)

= {1.2.3 + 2.3.4 + … + *k*(*k* + 1) (*k* + 2)} + (*k* + 1) (*k* + 2) (*k* + 3)

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 5:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*) :

For *n* = 1, we have

P(1): 1.3 = 3, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1.3 + 2.3^{2} + 3.3^{3} + … + *k*3^{k}+ (*k* + 1) 3^{k}^{+1}

= (1.3 + 2.3^{2} + 3.3^{3} + …+ *k.*3^{k}) + (*k* + 1) 3^{k}^{+1}

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 6:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): , which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

1.2 + 2.3 + 3.4 + … + *k*.(*k *+ 1) + (*k* + 1).(*k* + 2)

= [1.2 + 2.3 + 3.4 + … + *k*.(*k* + 1)] + (*k* + 1).(*k* + 2)

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 7:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

(1.3 + 3.5 + 5.7 + … + (2*k* – 1) (2*k* + 1) + {2(*k* + 1) – 1}{2(*k* + 1) + 1}

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 8:**

Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: 1.2 + 2.2^{2} + 3.2^{2} + … + *n*.2^{n} = (*n* – 1) 2^{n}^{+1} + 2

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): 1.2 + 2.2^{2} + 3.2^{2} + … + *n*.2^{n} = (*n* – 1) 2^{n}^{+1} + 2

For *n* = 1, we have

P(1): 1.2 = 2 = (1 – 1) 2^{1+1} + 2 = 0 + 2 = 2, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

1.2 + 2.2^{2} + 3.2^{2} + … + *k.*2^{k} = (*k* – 1) 2^{k}^{ + 1} + 2 … (i)

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 9:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): , which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 10:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 11:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):

For* n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Page No 95:**

**Question 12:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

For *n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 13:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

For *n* = 1, we have

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 14:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

For *n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 15:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 16:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 17:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

For *n* = 1, we have

, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 18:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

It can be noted that P(*n*) is true for *n* = 1 since .

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Hence,

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 19:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): *n* (*n* + 1) (*n* + 5), which is a multiple of 3.

It can be noted that P(*n*) is true for *n* = 1 since 1 (1 + 1) (1 + 5) = 12, which is a multiple of 3.

Let P(*k*) be true for some positive integer *k*, i.e.,

*k* (*k* + 1) (*k* + 5) is a multiple of 3.

∴*k* (*k* + 1) (*k* + 5) = 3*m*, where *m* ∈ **N** … (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 20:**

Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: 10^{2}^{n}^{ – 1 }+ 1 is divisible by 11.

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): 10^{2}^{n}^{ – 1 }+ 1 is divisible by 11.

It can be observed that P(*n*) is true for *n* = 1 since P(1) = 10^{2.1 – 1 }+ 1 = 11, which is divisible by 11.

Let P(*k*) be true for some positive integer *k*, i.e.,

10^{2}^{k}^{ – 1 }+ 1 is divisible by 11.

∴10^{2}^{k}^{ – 1 }+ 1 = 11*m*, where *m* ∈ **N **… (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 21:**

Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: *x*^{2}^{n} – *y*^{2}^{n} is divisible by* x *+ *y*.

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): *x*^{2}^{n} – *y*^{2}^{n} is divisible by* x *+ *y*.

It can be observed that P(*n*) is true for *n* = 1.

This is so because *x*^{2 }^{×}^{ 1} – *y*^{2 }^{×}^{ 1} = *x*^{2} – *y*^{2} = (*x *+ *y*) (*x* – *y*) is divisible by (*x* + *y*).

Let P(*k*) be true for some positive integer *k*, i.e.,

*x*^{2}^{k} – *y*^{2}^{k} is divisible by* x *+ *y*.

∴*x*^{2}^{k} – *y*^{2}^{k} = *m* (*x *+ *y*), where *m* ∈ **N** … (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 22:**

Prove the following by using the principle of mathematical induction for all *n* ∈ *N*: 3^{2}^{n}^{ + 2} – 8*n* – 9 is divisible by 8.

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): 3^{2}^{n}^{ + 2} – 8*n* – 9 is divisible by 8.

It can be observed that P(*n*) is true for *n* = 1 since 3^{2 }^{×}^{ 1 + 2} – 8 × 1 – 9 = 64, which is divisible by 8.

Let P(*k*) be true for some positive integer *k*, i.e.,

3^{2}^{k}^{ + 2} – 8*k* – 9 is divisible by 8.

∴3^{2}^{k}^{ + 2} – 8*k* – 9 = 8*m*; where *m* ∈ **N** … (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 23:**

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

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*):41^{n} – 14^{n}is a multiple of 27.

It can be observed that P(*n*) is true for *n* = 1 since **, **which is a multiple of 27.

Let P(*k*) be true for some positive integer *k*, i.e.,

41^{k} – 14^{k}is a multiple of 27

∴41^{k} – 14^{k} = 27*m*, where *m* ∈ **N** … (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

**Question 24:**

Prove the following by using the principle of mathematical induction for all

(2*n *+7) < (*n* + 3)^{2}

**Answer:**

Let the given statement be P(*n*), i.e.,

P(*n*): (2*n *+7) < (*n* + 3)^{2}

It can be observed that P(*n*) is true for *n* = 1 since 2.1 + 7 = 9 < (1 + 3)^{2} = 16, which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

(2*k* + 7) < (*k* + 3)^{2} … (1)

We shall now prove that P(*k* + 1) is true whenever P(*k*) is true.

Consider

Thus, P(*k* + 1) is true whenever P(*k*) is true.

*n*) is true for all natural numbers i.e., *n*.

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