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

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 + 3² + …+ 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 + 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.

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): 1³ = 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

1³ + 2³ + 3³ + … + k³ + (k + 1)³

= (1³ + 2³ + 3³ + …. + k³) + (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.

Hence, by the principle of mathematical induction, statement P(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 + 2.3² + 3.3³ + … + k3^k+ (k + 1) 3^k+1

= (1.3 + 2.3² + 3.3³ + …+ k.3^k) + (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.

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.

Hence, by the principle of mathematical induction, statement P(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 + … + (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.

Question 8:

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

Answer:

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

P(n): 1.2 + 2.2² + 3.2² + … + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

For n = 1, we have

P(1): 1.2 = 2 = (1 – 1) 2¹⁺¹ + 2 = 0 + 2 = 2, which is true.

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

1.2 + 2.2² + 3.2² + … + 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.

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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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.

Hence, by the principle of mathematical induction, statement P(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) = 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.

Question 20:

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^{2k – 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.

Question 21:

Prove the following by using the principle of mathematical induction for all n ∈ N: x²ⁿ – y²ⁿ is divisible by x + y.

Answer:

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

P(n): x²ⁿ – y²ⁿ 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 = x² – y² = (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.

Question 22:

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 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^{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.

Question 23:

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

Answer:

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

P(n):41ⁿ – 14ⁿis 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^kis 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.

Question 24:

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

(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.