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Question 9 - Mathematical Induction - Questions and Solutions - Mathematical Induction

Last updated at Dec. 16, 2024 by Teachoo

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Question 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

  1. Mathematical Induction
  2. Serial order wise

    3. Mathematical Induction - Questions and Solutions

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Mathematical Induction
Serial order wise

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo