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Chapter 4 Class 11 Mathematical Induction
Ex 4.1, 11 Important Deleted for CBSE Board 2023 Exams
Ex 4.1, 13 Important Deleted for CBSE Board 2023 Exams
Ex 4.1, 21 Important Deleted for CBSE Board 2023 Exams
Ex 4.1, 23 Important Deleted for CBSE Board 2023 Exams
Ex 4.1, 1 Important Deleted for CBSE Board 2023 Exams
Chapter 4 Class 11 Mathematical Induction
Last updated at March 16, 2023 by Teachoo
Ex 4.1, 7: Prove the following by using the principle of mathematical induction for all n N: 1.3 + 3.5 + 5.7 + + (2n 1) (2n + 1) = ( (4 2 + 6 1))/3 Let P(n) : 1.3 + 3.5 + 5.7 + + (2n 1) (2n + 1) = ( (4 2 + 6 1))/3 For n = 1, L.H.S = 1.3 = 3 R.H.S = (1(4.12 + 6.1 1))/3 = (4 + 6 1)/3 = 9/3 = 3 L.H.S. = R.H.S P(n) is true for n = 1 Assume P(k) is true 1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) = ( (4 2 + 6 1))/3 We will prove that P(k + 1) is true. 1.3 + 3.5 + 5.7 + + (2(k + 1) 1).(2(k + 1) + 1) = ( + 1)(4( + 1)^2 + 6( + 1) 1 )/3 1.3 + 3.5 + 5.7 + + (2k + 2 1).(2k + 2 + 1) = ( + 1)(4( ^2 + 1 + 2 )+ 6 + 6 1)/3 1.3 + 3.5 + 5.7 + + (2k + 1).(2k + 3) = ( + 1)(4 ^2 +4(1) +4(2 ) + 6 + 6 1)/3 1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) + (2k + 1).(2k + 3) = ( + 1)(4 ^2 + 4 + 8 + 6 + 6 1)/3 = ( + 1)(4 ^2 +14 + 9)/3 = (( (4 ^2 +14 + 9)+ 1(4 ^2 +14 + 9)))/3 = ((4 ^3 +18 ^2 + 23 + 9))/3 Thus, P(k +1) :1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) + (2k + 1).(2k + 3) = ((4 ^3 +18 ^2 + 23 + 9))/3 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) = ( (4 2 + 6 1))/3 Adding (2k+1).(2k+3) both sides 1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) + (2k + 1).(2k + 3) = ( (4 2 + 6 1))/3 + (2k + 1).(2k + 3) = ( (4 2 + 6 1) + 3(2 + 1)(2 + 3))/3 = ( (4 2 + 6 1) + 3(2 (2 + 3) + 1(2 + 3)))/3 = ( (4 2 + 6 1) + 3(2 (2 ) +2 (3) + 2 + 3))/3 = ( (4 2 + 6 1) + 3(4 ^2+ 6 + 2 + 3))/3 = ( (4 2 + 6 1) + 3(4 ^2+8 + 3))/3 = ( (4 2 + 6 1) + (3(4 ^2 ) +3(8 ) + 3(3)))/3 = ( (4 2 + 6 1) + (12 ^2 + 24 + 9))/3 = (4 3 + 6 ^2 + (12 ^2 + 24 + 9))/3 = (4 3 + 6 ^2 + 12 ^2 + 24 + 9)/3 = ((4 ^3 +18 ^2 + 23 + 9))/3 Thus, 1.3 + 3.5 + 5.7 + + (2k 1) (2k + 1) + (2k + 1).(2k + 3) = ((4 ^3 +18 ^2 + 23 + 9))/3 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