Ex 4.1, 13 - Prove (1 + 3/1) (1 + 5/4) (1 + 7/9) .. + (1 + 2n+1) - Equal - Multiplication

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Ex 4.1,13 Prove the following by using the principle of mathematical induction for all n โˆˆ N: ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โ€ฆ.. ("1 + " ((2๐‘› + 1))/๐‘›2) = (n + 1)2 Let P(n) : ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โ€ฆ.. ("1 + " ((2๐‘› + 1))/๐‘›2) = (n + 1)2 For n = 1, L.H.S = ("1 + " 3/1) = 1 + 3 = 4 R.H.S = (1 + 1)2 = 22 = 4 Thus, L.H.S. = R.H.S , โˆดP(n) is true for n = 1 Assuming P(k) is true P(k) : ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โ€ฆ.. ("1 + " ((2๐‘˜ + 1))/๐‘˜2) = (k + 1)2 We will prove P(k + 1) is true R.H.S = ((k + 1) + 1)2 L.H.S = ("1 + " 3/1) ("1 + " 5/4) ("1 + " 7/9)โ€ฆ.. ("1 + " ((2(๐‘˜ + 1) + 1))/(๐‘˜ + 1)2) L.H.S = R.H.S โˆด P(k + 1) is true whenever P(k) is true. โˆด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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