Solution of Chapter 4. Principle of Mathematical Induction (Mathematics Book)

Exercise 4.1

1

Prove the following using the principle of mathematical induction for all n ∈ N
2

Prove the following using the principle of mathematical induction for all n ∈ N
3

Prove the following using the principle of mathematical induction for all n ∈ N
4

Prove the following using the principle of mathematical induction for all n ∈ N
5

Prove the following using the principle of mathematical induction for all n ∈ N
6

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

1.2 + 2.3 + 3.4 + …+n.(n+1) =

7

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

1.3 + 3.5 + 5.7 +…+(2n – 1)(2n + 1)=
8

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

1.2 + 2.22 + 3.23 + …+n.2n = (n – 1)2n + 1 + 2
9

Prove the following using the principle of mathematical induction for all n ∈ N
10

Prove the following using the principle of mathematical induction for all n ∈ N
11

Prove the following using the principle of mathematical induction for all n ∈ N
12

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

a + ar + ar2 + …+ arn–1 =

13

Prove the following using the principle of mathematical induction for all n ∈ N
14

Prove the following using the principle of mathematical induction for all n ∈ N
15

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

12 + 32 + 52 +…+(2n –1)2 =

16

Prove the following using the principle of mathematical induction for all n ∈ N
17

Prove the following using the principle of mathematical induction for all n ∈ N
18

Prove the following using the principle of mathematical induction for all n ∈ N
19

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

n (n + 1) (n + 5) is a multiple of 3.
20

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

102n – 1 + 1 is divisible by 11.
21

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

x2n – y2n is divisible by x + y.
22

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

32n + 2 – 8n – 9 is divisible by 8.
23

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

41n – 14n is a multiple of 27.
24

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

(2n + 7) < (n + 3)2.