Last updated at Feb. 15, 2020 by Teachoo

Transcript

Ex 4.1,8: Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.23 + … + n.2n = (n – 1) 2n+1 + 2 Let P(n): 1.2 + 2.22 + 3.23 + … + n.2n = (n – 1) 2n+1 + 2 For n = 1, L.H.S = 1.2 = 2 R.H.S = (1 – 1) 21+1 + 2 = 0 + 2 = 2, Hence, L.H.S. = R.H.S ∴ P(n) is true for n = 1 Assume P(k) is true 1.2 + 2.22 + 3.23 + … + k.2k = (k – 1) 2k + 1 + 2 We will prove that P(k + 1) is true. 1.2 + 2.22 + 3.23 + … + (k + 1).2k+1 = ((k + 1) – 1) 2 (k + 1) + 1 +2 1.2 + 2.22 + 3.23 + … + (k + 1).2k+1 = (k) 2 k + 2 + 2 1.2 + 2.22 + 3.23 + … + k.2k + (k+1).2k+1= (k) 2 k + 2 + 2 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.2 + 2.22 + 3.23 + … + k.2k = (k – 1) 2k + 1 + 2 Adding (k + 1).2k+1 both sides (1.2 + 2.22 + 3.23 +……..+k.2k) + (k+1) 2k+1 = (k – 1)2k+1 + 2 + (k + 1)2k+1 = k 2k+1 – 2k+1 + 2 + k 2k+1 + 2k+1 = k 2k+1 + k 2k+1 – 2k+1 + 2k+1 + 2 = 2 k 2k+1 + 2 = k (21 . 2k+1) + 2 = k (2k+2) + 2 = 2k+2 k + 2 Thus, (1.2 + 2.22 + 3.23 +……..+k.2k) + (k+1) 2k+1 = 2k+2 k + 2 which is the same as P(k + 1) ∴ P(k + 1) is true whenever P(k) is true. ∴ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

Ex 4.1

Ex 4.1, 1
Important
Deleted for CBSE Board 2022 Exams

Ex 4.1, 2 Deleted for CBSE Board 2022 Exams

Ex 4.1, 3 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 4 Deleted for CBSE Board 2022 Exams

Ex 4.1, 5 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 6 Deleted for CBSE Board 2022 Exams

Ex 4.1, 7 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 8 Important Deleted for CBSE Board 2022 Exams You are here

Ex 4.1, 9 Deleted for CBSE Board 2022 Exams

Ex 4.1, 10 Deleted for CBSE Board 2022 Exams

Ex 4.1, 11 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 12 Deleted for CBSE Board 2022 Exams

Ex 4.1, 13 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 14 Deleted for CBSE Board 2022 Exams

Ex 4.1, 15 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 16 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 17 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 18 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 19 Deleted for CBSE Board 2022 Exams

Ex 4.1, 20 Deleted for CBSE Board 2022 Exams

Ex 4.1, 21 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 22 Deleted for CBSE Board 2022 Exams

Ex 4.1, 23 Important Deleted for CBSE Board 2022 Exams

Ex 4.1, 24 Important Deleted for CBSE Board 2022 Exams

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.