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Example 3 - Prove 1/1.2 + 1/2.3 + 1/3.4 .. + 1/n(n + 1) = 1/n+1 - Equal - 1 upon addition

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Example 3 For all n โ‰ฅ 1, prove that 1/1.2 + 1/2.3 + 1/3.4 +โ€ฆโ€ฆ.+ 1/(๐‘›(๐‘› + 1)) = 1/(๐‘› + 1) Let P (n) : 1/1.2 + 1/2.3 + 1/3.4 +โ€ฆโ€ฆ.+ 1/(๐‘›(๐‘› + 1)) = 1/(๐‘› + 1) For n=1, L.H.S = 1/1.2 = 1/2 R.H.S = 1/(1+1) = 1/2 Hence, L.H.S. = R.H.S , โˆด P(n) is true for n = 1 Assume P(k) is true 1/1.2 + 1/2.3 + 1/3.4 +โ€ฆโ€ฆ.+ 1/(๐‘˜(๐‘˜+1)) = ๐‘˜/(๐‘˜+1) We will prove that P(k + 1) is true. R.H.S = ((k + 1))/(((k + 1)+ 1) ) L.H.S =1/1.2 + 1/2.3 + 1/3.4 +โ€ฆโ€ฆ.+ 1/((k + 1)((k + 1)+ 1)) โˆด By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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