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Ex 4.1, 12 - Prove a + ar + ar2 + ... + a rn-1 = a(rn - 1)/r-1 - Ex 4.1

Ex 4.1, 12 - Chapter 4 Class 11 Mathematical Induction - Part 2
Ex 4.1, 12 - Chapter 4 Class 11 Mathematical Induction - Part 3
Ex 4.1, 12 - Chapter 4 Class 11 Mathematical Induction - Part 4


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Ex 4.1, 12: Prove the following by using the principle of mathematical induction for all n ∈ N: a + ar + ar2 + ……..+ arn – 1 = (π‘Ž(π‘Ÿ^𝑛 βˆ’ 1))/(π‘Ÿ βˆ’ 1) Let P (n) : a + ar + ar2 + ……..+ arn – 1 = π‘Ž(π‘Ÿ^𝑛 βˆ’ 1)/(π‘Ÿ βˆ’ 1) For n = 1, L.H.S = a R.H.S = (π‘Ž(π‘Ÿ1 βˆ’ 1))/(π‘Ÿ βˆ’ 1) = (π‘Ž(π‘Ÿ βˆ’ 1))/(π‘Ÿ βˆ’ 1) = a L.H.S. = R.H.S ∴ P(n) is true for n = 1 Assume that P(k) is true a + ar + ar2 + ……..+ ark – 1 = π‘Ž(π‘Ÿ^π‘˜ βˆ’ 1)/(π‘Ÿ βˆ’ 1) We will prove that P(k + 1) is true. a + ar + ar2 + ……..+ ar(k + 1) – 1 = π‘Ž(π‘Ÿ^(π‘˜ + 1) βˆ’ 1)/(π‘Ÿ βˆ’ 1) a + ar + ar2 + ……..+ ark – 1 + ark = π‘Ž(π‘Ÿ^(π‘˜ + 1) βˆ’ 1)/(π‘Ÿ βˆ’ 1) We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) a + ar + ar2 + ……..+ ark – 1 = π‘Ž(π‘Ÿ^π‘˜ βˆ’ 1)/(π‘Ÿ βˆ’ 1) Adding ark both sides a + ar + ar2 + …….. +ark – 1 + ark = π‘Ž(π‘Ÿ^π‘˜ βˆ’ 1)/(π‘Ÿ βˆ’ 1) + ark = (π‘Ž(π‘Ÿ^π‘˜ βˆ’ 1) + (π‘Ÿ βˆ’ 1)π‘Žπ‘Ÿ^π‘˜)/(π‘Ÿ βˆ’ 1) = (π‘Žπ‘Ÿ^π‘˜ βˆ’ π‘Ž + π‘Žπ‘Ÿ^π‘˜ (π‘Ÿ) βˆ’ π‘Žπ‘Ÿ^π‘˜)/(π‘Ÿ βˆ’ 1) = (π‘Žπ‘Ÿ^π‘˜βˆ’ π‘Žπ‘Ÿ^π‘˜ βˆ’ π‘Ž + π‘Žπ‘Ÿ^π‘˜ (π‘Ÿ))/(π‘Ÿ βˆ’ 1) = (0 βˆ’ π‘Ž + π‘Žπ‘Ÿ^π‘˜ (π‘Ÿ))/(π‘Ÿ βˆ’ 1) = (βˆ’ π‘Ž + π‘Žπ‘Ÿ^π‘˜ (π‘Ÿ))/(π‘Ÿ βˆ’ 1) = (βˆ’ π‘Ž + π‘Žπ‘Ÿ^π‘˜ (π‘Ÿ^1 ))/(π‘Ÿ βˆ’ 1) = (βˆ’ π‘Ž + π‘Žπ‘Ÿ^(π‘˜ + 1))/(π‘Ÿ βˆ’ 1) = (π‘Ž (βˆ’1 + π‘Ÿ^(π‘˜ + 1) ))/(π‘Ÿ βˆ’ 1) = π‘Ž(π‘Ÿ^(π‘˜ + 1) βˆ’ 1)/(π‘Ÿ βˆ’ 1) Thus, a + ar + ar2 + ……..+ ark – 1 + ark = π‘Ž(π‘Ÿ^(π‘˜ + 1) βˆ’ 1)/(π‘Ÿ βˆ’ 1) which is the same as P(k + 1) ∴ P(k + 1) is true whenever P(k) is true. ∴ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.