**Exercise 4.1 : Solutions of Questions on Page Number : 94**

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction is an important study material for students who are pursuing science or mathematics in their senior secondary education. The chapter explores the basic principles and concepts of mathematical induction, and the solutions are structured in a concise and simple manner, making it easy for students to understand the concepts and score good marks in their exams. The NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction covers various topics such as the principle of mathematical induction, proving statements using induction, and many more.

**NCERT Solution for Class 11 Maths Chapter 4 Principle of Mathematical Induction**

**Q1 :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

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 + 32 + … + 3^{k-1} + 3^{(k+1) – 1}

= (1 + 3 + 32 +… + 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.

**Q2 :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.

**Q3 :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.

**Q4 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q5 :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 + 2.32 + 3.33 + … + k3k+ (k + 1) 3

^{k+1}

= (1.3 + 2.32 + 3.33 + …+ k.3k) + (k + 1) 3

^{k+1 }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.

**Q6 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q7 :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 + … + (2k -1) (2k + 1) + {2(k + 1) -1}{2(k + 1) + 1}

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.

**Q8 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q9 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q10 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q11 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q12 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q13 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q14 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q15 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q16 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q17 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q18 :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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q19 :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) = 3m, 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q20 :Prove the following by using the principle of mathematical induction for all n ∈ N: 10 ^{2n – 1} + 1 is divisible by 11.**

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

P(n): 10

^{2n 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

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

∴10

^{2-1}+ 1 = 11m, 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q21 :Prove the following by using the principle of mathematical induction for all n ∈ N: x ^{2n} – y^{2n} is divisible by x + y.**

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

P(n): x

^{2n}– y

^{2n}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

^{2k}– y

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

∴x

^{2k}– y

^{2k}= 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q22 :Prove the following by using the principle of mathematical induction for all n ∈ N: 3 ^{2n+2} – 8n – 9 is divisible by 8.**

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

P(n): 3

^{2n + 2}– 8n – 9 is divisible by 8.

It can be observed that P(n) is true for n = 1 since 32 ×1 + 2- 8 × 1 – 9 = 64, which is divisible by 8.

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

3

^{2k+ 2}– 8k – 9 is divisible by 8.

∴3

^{2k+ 2}– 8k – 9 = 8m; 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q23 :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}= 27m, 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

**Q24 :Prove the following by using the principle of mathematical induction for all n ∈ N****(2n +7) < (n + 3) ^{2}**

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

P(n): (2n +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.,

(2k + 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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.