Mathematical Induction
Serial order wise

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Transcript

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 Since, 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 Assume that P(k) is true, P(k): 1 + 2 + 3 + ……. + k = (𝑘(𝑘 + 1))/2 We will prove that P(k + 1) is true. P(k + 1): 1 + 2 + 3 +……. + (k + 1) = ((k + 1)( (k + 1) + 1))/2 P(k + 1): 1 + 2 + 3 +…….+ k + (k + 1) = ((𝐤 + 𝟏)(𝐤 + 𝟐))/𝟐 We have to prove P(k + 1) is true Solving LHS 1 + 2 + 3 +…….+ k + (k + 1) From (1): 1 + 2 + 3 + ……. + k = (𝑘(𝑘 + 1))/2 = (𝒌(𝒌 + 𝟏))/𝟐 + (k + 1) = (𝑘(𝑘 + 1) + 2(𝑘 + 1))/2 = ((𝒌 + 𝟏)(𝒌 + 𝟐))/𝟐 = RHS ∴ P(k + 1) is true when P(k) is true Step 4: Write the following line Thus, By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.