CBSE Class XI

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

10^2(k+1)−1 + 1

= 10^2k+2−1 + 1

=10^2k+1 + 1

=10²(10^2k−1 + 1 – 1) + 1

=10²(10^2k−1 + 1) – 10² + 1

=10² ∙ 11m – 100 + 1                [Using (1)]

= 100 × 11m – 99

=11(100m – 9)

=11r, where r = (100m – 9) is some natural number

Therefore, 10^{2(k+1) −1} + 1 is divisible by 11.

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

(k + 1){(k + 1) + 1} {(k+1) + 5}

=(k + 1)(k + 2) {(k + 5) +1}

=(k + 1)(k + 2) (k + 5) + (k + 1) (k + 2)

={k(k+1)(k + 5) + 2(k + 1) (k + 5)} + (k + 1) (k + 2)

=3m + (k + 1){2(k + 5) + 2(k + 2)}

=3m + (k + 1){2k + 10 + k + 2}

= 3m + 3(k + 1) (3k + 12)

=3m + 3(k + 1) (k + 4)

=3{m + (k + 1) (k + 4)} = 3 ×q, where q = {m + (k + 1) (k + 4)} is some natural number

Therefore, (k + 1) {(k + 1) + 1} {(k + 1) + 5} is a multiple of 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 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., 

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 17:

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

Answer

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

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 of all n ∈ N :

Answer

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

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

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

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

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

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

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

{1.2 + 2.2² + 3.2³ + …+k.2^k} + (k + 1)∙2^k+1

=(k – 1)2^k+1 + 2 + (k + 1)2^k+1

=2^k+1{(k – 1) + (k + 1)} + 2

=2^k+1.2k + 2

={(k + 1) – 1}2^(k+1)+1 + 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

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

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

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:

Answer

Question 3:

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

Answer

Question 2:

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

Answer

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 1:

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

Answer

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.