Ex 4.1, 18 - Prove 1 + 2 + 3 + .. + n < 1/8 (2n+1)2 - Induction - Ex 4.1

  1. Chapter 4 Class 11 Mathematical Induction
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Ex 4.1,18 Prove the following by using the principle of mathematical induction for all n ∈ N: 1 + 2 + 3 +…..+ n < 1/8 (2n+1)2 Let P (n) : 1 + 2 + 3 +…..+ n < 1/8 (2n+1)2 For n = 1 L.H.S = 1 R.H.S = 1/8 (2.1 + 1)2 = 1/8 ( 2 + 1)2 = 1/8 (3)2 = 9/8 Since 1 < 9/8 Thus L.H.S < R.H.S ∴ P(n) is true for n = 1 Assume P(k) is true 1 + 2 + 3 +…..+ k < 1/8 (2k+1)2 We will prove that P(k + 1) is true. L.H.S = 1 + 2 + 3 + ……. + (k + 1) R.H.S = 1/8 (2(k + 1) + 1)2

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