RD SHARMA SOLUTION CHAPTER- 12 Mathematical Induction I CLASS 11TH MATHEMATICS-EDUGROWN

Chapter 12 Mathematical Induction Exercise Ex. 12.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Chapter 12 Mathematical Induction Exercise Ex. 12.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

for all nNSolution 39

Question 42

Solution 42

Question 43

Solution 43

Question 29

Prove by the principle of mathematical induction

n³ – 7n + 3 is divisible by 3 for all n Î NSolution 29

Question 30

Prove by the principle of mathematical induction

1 + 2 + 2² +…. + 2ⁿ = 2^{n + 1} -1 for all n Î NSolution 30

Question 31

Prove by the principle of mathematical induction

Solution 31

Question 40

Prove that

cos a + cos (a + b) + cos (a + 2b) + …..+ cos (a + (n – 1)b)

Solution 40

Question 41

Solution 41

Question 44

Solution 44

Question 45

Prove that the number of subsets of a set containing n distinct elements is 2ⁿ for all n Î N.Solution 45

Question 46

A sequence a₁, a₂, a₃, …….. is defined by letting a₁ = 3 and a_k = 7 a_k-1 for all natural numbers k ³ 2. Show that a_n = 3.7^n-1 for all n Î N.Solution 46

Question 47

Solution 47

Question 48

A sequence x₀, x₁, x₂, x₃, ……. is defined by letting x₀ = 5 and x_k = 4 + x_k -1 for all natural number k. show that x_n = 5 + 4n for all n Î N using mathematical induction.Solution 48

Question 49

Using principle of mathematical induction prove that

Solution 49

Question 50

The distributive law from algebra states that for all real numbers c, a₁ and a₂, we have c (a₁ + a₂) = ca₁ + ca₂

Use this law and mathematical induction to prove that, for all natural numbers, n ³ 2, if c (a₁ + a₂ + …. + a_n) = ca₁ + ca₂ + …+ ca_n.Solution 50

Question 28

Solution 28

Question 35

Solution 35