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Ex 4.1, 9 - Prove 1/2 + 1/4 + 1/8 + ... + 1/2n = 1 - 1/2n - Ex 4.1

  1. Chapter 4 Class 11 Mathematical Induction
  2. Serial order wise
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Ex 4.1, 9: Prove the following by using the principle of mathematical induction for all n โˆˆ N: 1/2 + 1/4 + 1/8 + ....+ 1/2๐‘› = 1 โ€“ 1/2๐‘› Let P(n): 1/2 + 1/4 + 1/8 + ....+ 1/2๐‘› = 1 โ€“ 1/2๐‘› For n = 1, we have L.H.S = 1/2 R.H.S = 1 โ€“ 1/21 = 1/2 Hence, L.H.S. = R.H.S , โˆด P(n) is true for n = 1 Assume P(k) is true 1/2 + 1/4 + 1/8 + ....+ 1/2๐‘˜ = 1 โ€“ 1/2๐‘˜ We will prove that P(k + 1) is true. R.H.S = 1 โ€“ 1/2^(๐‘˜ + 1) L.H.S = 1/2 + 1/4 + 1/8 + ....+ 1/2^(๐‘˜ + 1) 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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