Ex 4.1, 3 - Prove by induction 1 + 1/(1 + 2) + 1/(1 + 2 + 3) + .. - Equal - 1 upon addition

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

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