Prove 1 + 2 + 3 ... + n = n(n+1)/2 - Mathematical Induction - Theory

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Prove 1 + 2 + 3 + ……. + n = (n(n+1))/2 for n, n is a natural number Step 1: Let P(n) : (the given statement)\ Let P(n): 1 + 2 + 3 + ……. + n = (n(n+1))/2 Step 2: Prove for n = 1 For n = 1, L.H.S = 1 R.H.S = (n(n+1))/2 = (1(1+1))/2 = (1 × (2))/2 = 1 L.H.S = R.H.S ∴ P(n) is true for n = 1 Step 3: Assume P(k) to be true and then prove P(k+1) is true Let P(k): 1 + 2 + 3 + ……. + k = (k(k+1))/2 be true We will prove P(k+1) is true, 1 + 2 + 3 +……. + (k+ 1) = ((k+1)( (k+1)+1))/2 1 + 2 + 3 +…….+ k + (k+ 1) = ((k+1)(k+2))/2 We will prove (2) from (1) From (1) 1 + 2 + 3 + ……. + k = (k(k+1))/2 Adding k + 1 both sides 1 + 2 + 3 +…….+ k + (k+ 1) = (k(k+1))/2 + (k+1) 1 + 2 + 3 +…….+ k + (k+ 1) = (k(k+1)+2(𝑘+1))/2 1 + 2 + 3 +…….+ k + (k+ 1) = ((k+1)(k+2))/2 Which is the same as P(k + 1) ∴ P(k+1) is true when P(k) is true Step 4: Write the following line ∴ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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