RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 4. Principle of Mathematical Induction

Subject: Maths - Class 11th

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Exercise 4

1

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

1 + 2 + 3 + 4 + … + n = 1/2 n(n + 1)
2

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

2 + 4 + 6 + 8 + …. + 2n = n(n + 1)
3

Using the principle of mathematical induction, prove each of the following for all n ϵ N:
4

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

2 + 6 + 18 + … + 2 × 3n–1 = (3n –1)
5

Using the principle of mathematical induction, prove each of the following for all n ϵ N:
6

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

12 + 32 + 52 + 72 + … + (2n – 1)2 =


7

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

1.2 + 2.22 + 3.23 + …. + n.2n =(n – 1)2n+1 + 2.
8

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

3.22 + 32.23 + 33.24 + …. + 3n.2n+1 = (6n – 1).
9

Using the principle of mathematical induction, prove each of the following for all n ϵ N:
10

Using the principle of mathematical induction, prove each of the following for all n ϵ N:
11

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

.
12

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

.
13

Using the principle of mathematical induction, prove each of the following for all n ϵ N:
14

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

= (n + 1)2.
15

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

= (n + 1).
16

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

n × ( n + 1 ) × ( n + 2 ) is multiple of 6
17

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(x2n – y2n) is divisible by (x + y).
18

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(x2n – 1) - 1 is divisible by (x – y), where x ≠ 1.
19

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

{(41)n – (14)n} is divisible by 27.
20

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(4n + 15n – 1) is divisible by 9.
21

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(32n+2 – 8n – 9) is divisible by 8.
22

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(23n – 1) is a multiple of 7
23

Using the principle of mathematical induction, prove each of the following for all n ϵ N:

3n≥ 2n.