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Mathematical Induction - Questions and Solutions

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Mathematical Induction

Serial order wise

Last updated at May 29, 2023 by Teachoo

Question11 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)