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

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Ex 4.1,17 Prove the following by using the principle of mathematical induction for all n โˆˆ N: 1/3.5 + 1/5.7 + 1/7.9 +โ€ฆโ€ฆ.+ 1/((2๐‘› + 1)(2๐‘› + 3)) = ๐‘›/(3(2๐‘› + 3)) Let P (n) : 1/3.5 + 1/5.7 + 1/7.9 +โ€ฆโ€ฆ.+ 1/((2๐‘› + 1)(2๐‘› + 3)) = ๐‘›/(3(2๐‘› + 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 ร— 5) = 1/15 Hence, L.H.S. = R.H.S , โˆด P(n) is true for n = 1 Assume P(k) is true 1/3.5 + 1/5.7 + 1/7.9 +โ€ฆโ€ฆ.+ 1/((2๐‘˜ + 1)(2๐‘˜ + 3)) = ๐‘˜/(3(2๐‘˜ + 3)) We will prove that P(k + 1) is true. R.H.S = ((๐‘˜ + 1))/3(2(๐‘˜ + 1) + 3) L.H.S = 1/3.5 + 1/5.7 + 1/7.9 +โ€ฆโ€ฆ.+ 1/((2(๐‘˜ + 1) + 1)(2(๐‘˜ + 1) + 3))

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