Ex 4.1, 6 - 1.2 + 2.3 + 3.4 + .. + n.(n+1) = n(n+1)(n+2)/3 - Equal - Addition

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Ex 4.1,6: Prove the following by using the principle of mathematical induction for all n โˆˆ N: 1.2 + 2.3 + 3.4 +โ€ฆโ€ฆ + n. (n + 1) = (๐‘›(๐‘› + 1)(๐‘› + 2))/3 Let P(n): 1.2 + 2.3 + 3.4 +โ€ฆ.+ n.(n + 1) = (๐‘›(๐‘› + 1)(๐‘› + 2))/3 For n = 1, L.H.S = 1.2 = 2 R.H.S = (1(1+1)(1+2))/3 = 1.2.3/3 = 2 L.H.S. = R.H.S โˆด P(n) is true for n = 1 Assume P(k) be true 1.2 + 2.3 + 3.4 +โ€ฆ.+ k.(k + 1) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2))/3 We will prove that P(k + 1) is true. 1.2 + 2.3 + 3.4 +โ€ฆ.+ (k + 1).((k + 1) + 1) = ((๐‘˜ + 1)((๐‘˜ + 1)+ 1)((๐‘˜ + 1)+ 2))/3 1.2 + 2.3 + 3.4 +โ€ฆ.+ (k + 1).(k + 2) = (๐‘˜ + 1)(๐‘˜ +2)(๐‘˜ +3)/3 1.2 + 2.3 + 3.4 +โ€ฆ. + k.(k + 1) + (k + 1).(k + 2) = (๐‘˜ + 1)(๐‘˜ +2)(๐‘˜ +3)/3 We have to prove P(k+1) from P(k) From (1) 1.2 + 2.3 + 3.4 +โ€ฆ.+ k.(k + 1) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2))/3 Adding (k + 1). (k + 2) both sides 1.2 + 2.3 + 3.4 +โ€ฆ.+ k.(k + 1) + (k + 1).(k + 2) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2))/3 + (k + 1).(k + 2) = (๐‘˜(๐‘˜ + 1)(๐‘˜ + 2)+3(๐‘˜ + 1)(๐‘˜ + 2))/3 = ((๐‘˜ + 1)(๐‘˜ + 2) (๐‘˜ + 3) )/3 Thus, 1.2 + 2.3 + 3.4 +โ€ฆ. + k.(k + 1) + (k + 1).(k + 2) = (๐‘˜ + 1)(๐‘˜ +2)(๐‘˜ +3)/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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