Ex 4.1, 7 - Prove 1.3 + 3.5 + 5.7 + .. + (2n-1) (2n+1) - Class 11 - Equal - Addition

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Ex 4.1, 7: Prove the following by using the principle of mathematical induction for all n โˆˆ N: 1.3 + 3.5 + 5.7 + โ€ฆ + (2n โ€“ 1) (2n + 1) = (๐‘›(4๐‘›2 + 6๐‘› โˆ’ 1))/3 Let P(n) : 1.3 + 3.5 + 5.7 + โ€ฆ + (2n โ€“ 1) (2n + 1) = (๐‘›(4๐‘›2 + 6๐‘› โˆ’ 1))/3 For n = 1, L.H.S = 1.3 = 3 R.H.S = (1(4.12 + 6.1 โˆ’ 1))/3 = (4 + 6 โˆ’ 1)/3 = 9/3 = 3 L.H.S. = R.H.S โˆด P(n) is true for n = 1 Assume P(k) is true 1.3 + 3.5 + 5.7 + โ€ฆ + (2k โ€“ 1) (2k + 1) = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1))/3 We will prove that P(k + 1) is true. 1.3 + 3.5 + 5.7 + โ€ฆ + (2(k + 1) โ€“ 1).(2(k + 1) + 1) = (๐‘˜ + 1)(4(๐‘˜ + 1)^2 + 6(๐‘˜ + 1) โˆ’ 1 )/3 1.3 + 3.5 + 5.7 + โ€ฆ + (2k + 2 โ€“ 1).(2k + 2 + 1) = (๐‘˜ + 1)(4(๐‘˜^2 + 1 + 2๐‘˜)+ 6๐‘˜ + 6 โˆ’ 1)/3 1.3 + 3.5 + 5.7 + โ€ฆ + (2k + 1).(2k + 3) = (๐‘˜ + 1)(4๐‘˜^2 +4(1) +4(2๐‘˜) + 6๐‘˜ + 6 โˆ’ 1)/3 1.3 + 3.5 + 5.7 + โ€ฆ + (2k โ€“ 1) (2k + 1) + (2k + 1).(2k + 3) = (๐‘˜ + 1)(4๐‘˜^2 + 4 + 8๐‘˜ + 6๐‘˜ + 6 โˆ’ 1)/3 = (๐‘˜ + 1)(4๐‘˜^2 +14๐‘˜ + 9)/3 = ((๐‘˜(4๐‘˜^2 +14๐‘˜ + 9)+ 1(4๐‘˜^2 +14๐‘˜ + 9)))/3 = ((4๐‘˜^3 +18๐‘˜^2 + 23๐‘˜ + 9))/3 Thus, P(k +1) :1.3 + 3.5 + 5.7 + โ€ฆ + (2k โ€“ 1) (2k + 1) + (2k + 1).(2k + 3) = ((4๐‘˜^3 +18๐‘˜^2 + 23๐‘˜ + 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 + โ€ฆ + (2k โ€“ 1) (2k + 1) = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1))/3 Adding (2k+1).(2k+3) both sides 1.3 + 3.5 + 5.7 + โ€ฆ + (2k โ€“ 1) (2k + 1) + (2k + 1).(2k + 3) = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1))/3 + (2k + 1).(2k + 3) = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + 3(2๐‘˜ + 1)(2๐‘˜ + 3))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + 3(2๐‘˜(2๐‘˜ + 3) + 1(2๐‘˜ + 3)))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + 3(2๐‘˜(2๐‘˜) +2๐‘˜(3) + 2๐‘˜ + 3))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + 3(4๐‘˜^2+ 6๐‘˜ + 2๐‘˜ + 3))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + 3(4๐‘˜^2+8๐‘˜ + 3))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + (3(4๐‘˜^2 ) +3(8๐‘˜) + 3(3)))/3 = (๐‘˜(4๐‘˜2 + 6๐‘˜ โˆ’ 1) + (12๐‘˜^2 + 24๐‘˜ + 9))/3 = (4๐‘˜3 + 6๐‘˜^2 โˆ’ ๐‘˜ + (12๐‘˜^2 + 24๐‘˜ + 9))/3 = (4๐‘˜3 + 6๐‘˜^2 + 12๐‘˜^2 โˆ’ ๐‘˜ + 24๐‘˜ + 9)/3 = ((4๐‘˜^3 +18๐‘˜^2 + 23๐‘˜ + 9))/3 Thus, 1.3 + 3.5 + 5.7 + โ€ฆ + (2k โ€“ 1) (2k + 1) + (2k + 1).(2k + 3) = ((4๐‘˜^3 +18๐‘˜^2 + 23๐‘˜ + 9))/3 which is the same as P(k +1) โˆด P(k + 1) is true whenever P(k) is true. โˆด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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