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

#### Ex 4.1

#### Page No 94:

#### 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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