Ex 4.1, 17

Last updated at Dec. 8, 2016 by Teachoo

Ex 4.1, 17 - Prove 1/3.5 + 1/5.7 + 1/7.9 .. + 1/(2n + 1)(2n + 3) - Equal - 1 upon addition

Transcript

Ex 4.1,17Prove the following by using the principle of mathematical induction for all n ∈ N: 1/3.5 + 1/5.7 + 1/7.9 +…….+ 1/((2𝑛 + 1)(2𝑛 + 3)) = 𝑛/(3(2𝑛 + 3)) Let P (n) : 1/3.5 + 1/5.7 + 1/7.9 +…….+ 1/((2𝑛 + 1)(2𝑛 + 3)) = 𝑛/(3(2𝑛 + 3)) For n = 1, L.H.S = 1/3.5 = 1/15 R.H.S = 1/(3(2(1) + 3)) = 1/(3(2 +3)) = 1/(3 × 5) = 1/15 Hence, L.H.S. = R.H.S , ∴ P(n) is true for n = 1 Assume P(k) is true 1/3.5 + 1/5.7 + 1/7.9 +…….+ 1/((2𝑘 + 1)(2𝑘 + 3)) = 𝑘/(3(2𝑘 + 3)) We will prove that P(k + 1) is true. R.H.S = ((𝑘 + 1))/3(2(𝑘 + 1) + 3) L.H.S = 1/3.5 + 1/5.7 + 1/7.9 +…….+ 1/((2(𝑘 + 1) + 1)(2(𝑘 + 1) + 3))

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