**Ex 4.1, 7**

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 4.1, 7: Prove the following by using the principle of mathematical induction for all n β N: 1.3 + 3.5 + 5.7 + β¦ + (2n β 1) (2n + 1) = (π(4π2 + 6π β 1))/3 Let P(n) : 1.3 + 3.5 + 5.7 + β¦ + (2n β 1) (2n + 1) = (π(4π2 + 6π β 1))/3 For n = 1, L.H.S = 1.3 = 3 R.H.S = (1(4.12 + 6.1 β 1))/3 = (4 + 6 β 1)/3 = 9/3 = 3 L.H.S. = R.H.S β΄ P(n) is true for n = 1 Assume P(k) is true 1.3 + 3.5 + 5.7 + β¦ + (2k β 1) (2k + 1) = (π(4π2 + 6π β 1))/3 We will prove that P(k + 1) is true. 1.3 + 3.5 + 5.7 + β¦ + (2(k + 1) β 1).(2(k + 1) + 1) = (π + 1)(4(π + 1)^2 + 6(π + 1) β 1 )/3 1.3 + 3.5 + 5.7 + β¦ + (2k + 2 β 1).(2k + 2 + 1) = (π + 1)(4(π^2 + 1 + 2π)+ 6π + 6 β 1)/3 1.3 + 3.5 + 5.7 + β¦ + (2k + 1).(2k + 3) = (π + 1)(4π^2 +4(1) +4(2π) + 6π + 6 β 1)/3 1.3 + 3.5 + 5.7 + β¦ + (2k β 1) (2k + 1) + (2k + 1).(2k + 3) = (π + 1)(4π^2 + 4 + 8π + 6π + 6 β 1)/3 = (π + 1)(4π^2 +14π + 9)/3 = ((π(4π^2 +14π + 9)+ 1(4π^2 +14π + 9)))/3 = ((4π^3 +18π^2 + 23π + 9))/3 Thus, P(k +1) :1.3 + 3.5 + 5.7 + β¦ + (2k β 1) (2k + 1) + (2k + 1).(2k + 3) = ((4π^3 +18π^2 + 23π + 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 + β¦ + (2k β 1) (2k + 1) = (π(4π2 + 6π β 1))/3 Adding (2k+1).(2k+3) both sides 1.3 + 3.5 + 5.7 + β¦ + (2k β 1) (2k + 1) + (2k + 1).(2k + 3) = (π(4π2 + 6π β 1))/3 + (2k + 1).(2k + 3) = (π(4π2 + 6π β 1) + 3(2π + 1)(2π + 3))/3 = (π(4π2 + 6π β 1) + 3(2π(2π + 3) + 1(2π + 3)))/3 = (π(4π2 + 6π β 1) + 3(2π(2π) +2π(3) + 2π + 3))/3 = (π(4π2 + 6π β 1) + 3(4π^2+ 6π + 2π + 3))/3 = (π(4π2 + 6π β 1) + 3(4π^2+8π + 3))/3 = (π(4π2 + 6π β 1) + (3(4π^2 ) +3(8π) + 3(3)))/3 = (π(4π2 + 6π β 1) + (12π^2 + 24π + 9))/3 = (4π3 + 6π^2 β π + (12π^2 + 24π + 9))/3 = (4π3 + 6π^2 + 12π^2 β π + 24π + 9)/3 = ((4π^3 +18π^2 + 23π + 9))/3 Thus, 1.3 + 3.5 + 5.7 + β¦ + (2k β 1) (2k + 1) + (2k + 1).(2k + 3) = ((4π^3 +18π^2 + 23π + 9))/3 which is the same as P(k +1) β΄ P(k + 1) is true whenever P(k) is true. β΄ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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Davneet Singh

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