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

Transcript

Ex 4.1, 7: Prove the following by using the principle of mathematical induction for all n โ N: 1.3 + 3.5 + 5.7 + โฆ + (2n โ 1) (2n + 1) = (๐(4๐2 + 6๐ โ 1))/3 Let P(n) : 1.3 + 3.5 + 5.7 + โฆ + (2n โ 1) (2n + 1) = (๐(4๐2 + 6๐ โ 1))/3 For n = 1, L.H.S = 1.3 = 3 R.H.S = (1(4.12 + 6.1 โ 1))/3 = (4 + 6 โ 1)/3 = 9/3 = 3 L.H.S. = R.H.S โด 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 โ 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) โ 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 โ 1)/3 1.3 + 3.5 + 5.7 + โฆ + (2k + 1).(2k + 3) = (๐ + 1)(4๐^2 +4(1) +4(2๐) + 6๐ + 6 โ 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 โ 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 โ 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 โ 1))/3 + (2k + 1).(2k + 3) = (๐(4๐2 + 6๐ โ 1) + 3(2๐ + 1)(2๐ + 3))/3 = (๐(4๐2 + 6๐ โ 1) + 3(2๐(2๐ + 3) + 1(2๐ + 3)))/3 = (๐(4๐2 + 6๐ โ 1) + 3(2๐(2๐) +2๐(3) + 2๐ + 3))/3 = (๐(4๐2 + 6๐ โ 1) + 3(4๐^2+ 6๐ + 2๐ + 3))/3 = (๐(4๐2 + 6๐ โ 1) + 3(4๐^2+8๐ + 3))/3 = (๐(4๐2 + 6๐ โ 1) + (3(4๐^2 ) +3(8๐) + 3(3)))/3 = (๐(4๐2 + 6๐ โ 1) + (12๐^2 + 24๐ + 9))/3 = (4๐3 + 6๐^2 โ ๐ + (12๐^2 + 24๐ + 9))/3 = (4๐3 + 6๐^2 + 12๐^2 โ 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) โด P(k + 1) is true whenever P(k) is true. โด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

Serial order wise