Chapter 4 Class 11 Mathematical Induction
Question 11 Important Deleted for CBSE Board 2025 Exams
Question 13 Important Deleted for CBSE Board 2025 Exams
Question 21 Important Deleted for CBSE Board 2025 Exams
Question 23 Important Deleted for CBSE Board 2025 Exams
Question 1 Important Deleted for CBSE Board 2025 Exams
Chapter 4 Class 11 Mathematical Induction
Last updated at April 16, 2024 by Teachoo
Question 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