Slide38.JPG

Slide39.JPG
Slide40.JPG
Slide41.JPG

  1. Chapter 4 Class 11 Mathematical Induction (Deleted)
  2. Serial order wise

Transcript

Ex 4.1, 1 Prove the following by using the principle of mathematical induction for all n ∈ N: 1 + 3 + 32+……+ 3n – 1 = ((3𝑛 βˆ’ 1))/2 Let P(n) : 1 + 3 + 32+……+ 3n – 1 = ((3𝑛 βˆ’ 1))/2 Proving for n = 1 For n = 1, L.H.S = 1 R.H.S = ((3^1 βˆ’ 1))/2 = ((3 βˆ’ 1))/2 = ((2))/2 = 1 Since, L.H.S. = R.H.S ∴ P(n) is true for n = 1 Proving P(k + 1) is true if P(k) is true Assume that P(k) is true, P(k): 1 + 3 + 32 +…..+ 3k – 1 = ((3π‘˜ βˆ’ 1))/2 We will prove that P(k + 1) is true. P(k + 1): 1 + 3 + 32 +…..+ 3(k + 1) – 1 = ((3^(π‘˜+1) βˆ’ 1))/2 P(k + 1): 1 + 3 + 32 +…..3(k – 1) + 3(k) = ((3^(π‘˜+1) βˆ’ 1))/2 We have to prove P(k + 1) is true Solving LHS 1 + 3 + 32 +…..+ 3k – 1 + 3k From (1): 1 + 3 + 32 +…..+ 3k – 1 = ((πŸ‘π’Œ βˆ’ 𝟏))/𝟐 = ((πŸ‘π’Œ βˆ’ 𝟏))/𝟐 + 3k = ((3π‘˜ βˆ’ 1) + 2 Γ— 3^π‘˜)/2 = (πŸ‘π’Œ + 𝟐 Γ— πŸ‘^π’Œ βˆ’ 𝟏)/𝟐 = ( 3(3^π‘˜ )βˆ’ 1)/2 = (πŸ‘^(π’Œ + 𝟏) βˆ’ 𝟏)/𝟐 = RHS ∴ P(k + 1) is true when P(k) is true Thus, By the principle of mathematical induction, P(n) is true for n, where n is a natural number

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
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.