# NCERT Solutions for Class 11 Maths Maths Chapter 4

## Principle of Mathematical Induction Class 11

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

Q1 :

Prove the following by using the principle of mathematical induction for all nN: Let the given statement be P(n), i.e.,

P(n): 1 + 3 + 32 + …+ 3nâ€“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 + … + 3kâ€“1 + 3(k+1) â€“ 1

= (1 + 3 + 32 +… + 3kâ€“1) + 3k 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 nN: 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.

Q3 :

Prove the following by using the principle of mathematical induction for all nN: Q4 :

Prove the following by using the principle of mathematical induction for all nN: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) = Q5 :

Prove the following by using the principle of mathematical induction for all nN: Q6 :

Prove the following by using the principle of mathematical induction for all nN: Q7 :

Prove the following by using the principle of mathematical induction for all nN: Q8 :

Prove the following by using the principle of mathematical induction for all nN: 1.2 + 2.22 + 3.22 + … + n.2n = (n - 1) 2n+1 + 2

Q9 :

Prove the following by using the principle of mathematical induction for all nN: Q10 :

Prove the following by using the principle of mathematical induction for all nN: Q11 :

Prove the following by using the principle of mathematical induction for all nN: Q12 :

Prove the following by using the principle of mathematical induction for all nN: Q13 :

Prove the following by using the principle of mathematical induction for all nN: Q14 :

Prove the following by using the principle of mathematical induction for all nN: Q15 :

Prove the following by using the principle of mathematical induction for all nN: Q16 :

Prove the following by using the principle of mathematical induction for all nN: Q17 :

Prove the following by using the principle of mathematical induction for all nN: Q18 :

Prove the following by using the principle of mathematical induction for all nN: Q19 :

Prove the following by using the principle of mathematical induction for all nN: n (n + 1) (n + 5) is a multiple of 3.

Q20 :

Prove the following by using the principle of mathematical induction for all nN: 102n - 1 + 1 is divisible by 11.

Q21 :

Prove the following by using the principle of mathematical induction for all nN: x2n - y2n is divisible by x + y.

Q22 :

Prove the following by using the principle of mathematical induction for all nN: 32n + 2 - 8n - 9 is divisible by 8.

Q23 :

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

Q24 :

Prove the following by using the principle of mathematical induction for all (2n +7) < (n + 3)2