NCERT Solutions Class 11 Maths Chapter 4 - Principle Of Mathematical Induction

NCERT QUESTION - (Principle Of Mathematical Induction)

Question 1:  Prove the following by using the principle of mathematical induction or 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) 2^{n+ 1} + 2
ANSWER:  Let the given statement be P(n), i.e.,

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

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 – 14ⁿ 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.

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.