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Mathematical Induction
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Ex 4.1, 4 - Prove 1.2.3 + 2.3.4 + .. + n(n + 1) (n + 2) = n(n+1) - Equal - Addition

Ex 4.1, 4 - Chapter 4 Class 11 Mathematical Induction - Part 2
Ex 4.1, 4 - Chapter 4 Class 11 Mathematical Induction - Part 3

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Question 4: Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + â€Ķ + n(n + 1) (n + 2) = (𝑛(𝑛 + 1)(𝑛 + 2)(𝑛 + 3))/4 Let P(n): 1.2.3 + 2.3.4 + â€Ķ + n(n + 1) (n + 2) = (𝑛(𝑛 + 1)(𝑛 + 2)(𝑛 + 3))/4 For n = 1, L.H.S = 1.2.3 = 6 R.H.S = (1(1+1)(1+2)(1+3))/4 = (1 × 2 × 3 × 4)/4 = 6 L.H.S. = R.H.S âˆīP(n) is true for n = 1 Assume P(k) is true 1.2.3 + 2.3.4 + â€Ķ + k(k + 1) (k + 2) = (𝑘(𝑘 + 1)(𝑘 + 2)(𝑘 + 3))/4 We will prove that P(k + 1) is true. 1.2.3 + 2.3.4 + â€Ķ + (k + 1) ((k +1) + 1) ((k + 1) + 2) = (𝑘 + 1)((𝑘 + 1)+ 1)((𝑘 + 1)+ 2)((𝑘 + 1)+ 3)/4 1.2.3 + 2.3.4 + â€Ķ + (k + 1) (k + 2) (k + 3) = ((𝑘 + 1)(𝑘 + 2)(𝑘 + 3)(𝑘 + 4))/4 1.2.3 + 2.3.4 + â€Ķ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((𝑘 + 1)(𝑘 + 2)(𝑘 + 3)(𝑘 + 4))/4 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.2.3 + 2.3.4 + â€Ķ + k(k + 1) (k + 2) = (𝑘(𝑘 + 1)(𝑘 + 2)(𝑘 + 3))/4 Adding (k+1) (k+2) (k+3) both sides 1.2.3 + 2.3.4 + â€Ķ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = (𝑘(𝑘 + 1)(𝑘 + 2)(𝑘 + 3))/4 +(k + 1) (k + 2) (k + 3) = (𝑘(𝑘 + 1)(𝑘 + 2)(𝑘 + 3) + 4(𝑘 + 1)(𝑘 + 2)(𝑘 + 3))/4 = ((𝑘 + 1)(𝑘 + 2)(𝑘 + 3) (𝑘 + 4))/4 Thus, 1.2.3 + 2.3.4 + â€Ķ + k(k + 1) (k + 2) +(k + 1) (k + 2) (k + 3) = ((𝑘 + 1)(𝑘 + 2)(𝑘 + 3)(𝑘 + 4))/4 which is the same as P(k + 1) âˆī P(k + 1) is true whenever P(k) is true. âˆī 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.