# Ex 4.1, 11 - Chapter 4 Class 11 Mathematical Induction

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 4.1,11 Prove the following by using the principle of mathematical induction for all n ∈ N: 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑛 𝑛 + 1(𝑛 + 2) = 𝑛(𝑛 + 3)4(𝑛 + 1)(𝑛 + 2) Let P (n) : 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑛 𝑛 + 1(𝑛 + 2) = 𝑛(𝑛 + 3)4(𝑛 + 1)(𝑛 + 2) For n = 1, L.H.S = 11.2.3 = 16 R.H.S = 1.(1 + 3)4(1 + 1)(1 + 2) = 1.44.2.3 = 12.3 = 16 Hence, L.H.S. = R.H.S , ∴ P(n) is true for n = 1 Assume P(k) is true 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1𝑘 𝑘 + 1(𝑘 + 2) = 𝑘(𝑘 + 3)4(𝑘 + 1)(𝑘 + 2) We will prove that P(k + 1) is true. R.H.S = 𝑘 + 1 𝑘 + 1+ 34 𝑘 + 1+ 1 𝑘 + 1+ 2 L.H.S = 11.2.3 + 12.3.4 + 13.4.5 + ……..+ 1 𝑘 + 1 𝑘 + 1+ 1( 𝑘 + 1+ 2) Rough Factorizing 𝐤𝟑+ 𝟒+ 𝟗𝐤+ 𝟔𝐤𝟐 Let f(k) = k3+ 4+ 9k+ 6k2 Putting k = − 1 f(−1) = (−1)3+ 4+9 −1+6 (−1)2 = −1 + 4− 9+6 = 0 Thus, (k + 1) is a factor of f(k) ∴ f(k) = (𝑘+1)( 𝑘2+5𝑘+4)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.