Mathematical Induction
Serial order wise

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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Transcript

Question 12: Prove the following by using the principle of mathematical induction for all n ∈ N: a + ar + ar2 + ……..+ arn – 1 = (𝑎(𝑟^𝑛 − 1))/(𝑟 − 1) Let P (n) : a + ar + ar2 + ……..+ arn – 1 = 𝑎(𝑟^𝑛 − 1)/(𝑟 − 1) For n = 1, L.H.S = a R.H.S = (𝑎(𝑟1 − 1))/(𝑟 − 1) = (𝑎(𝑟 − 1))/(𝑟 − 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 = 𝑎(𝑟^𝑘 − 1)/(𝑟 − 1) We will prove that P(k + 1) is true. a + ar + ar2 + ……..+ ar(k + 1) – 1 = 𝑎(𝑟^(𝑘 + 1) − 1)/(𝑟 − 1) a + ar + ar2 + ……..+ ark – 1 + ark = 𝑎(𝑟^(𝑘 + 1) − 1)/(𝑟 − 1) We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) a + ar + ar2 + ……..+ ark – 1 = 𝑎(𝑟^𝑘 − 1)/(𝑟 − 1) Adding ark both sides a + ar + ar2 + …….. +ark – 1 + ark = 𝑎(𝑟^𝑘 − 1)/(𝑟 − 1) + ark = (𝑎(𝑟^𝑘 − 1) + (𝑟 − 1)𝑎𝑟^𝑘)/(𝑟 − 1) = (𝑎𝑟^𝑘 − 𝑎 + 𝑎𝑟^𝑘 (𝑟) − 𝑎𝑟^𝑘)/(𝑟 − 1) = (𝑎𝑟^𝑘− 𝑎𝑟^𝑘 − 𝑎 + 𝑎𝑟^𝑘 (𝑟))/(𝑟 − 1) = (0 − 𝑎 + 𝑎𝑟^𝑘 (𝑟))/(𝑟 − 1) = (− 𝑎 + 𝑎𝑟^𝑘 (𝑟))/(𝑟 − 1) = (− 𝑎 + 𝑎𝑟^𝑘 (𝑟^1 ))/(𝑟 − 1) = (− 𝑎 + 𝑎𝑟^(𝑘 + 1))/(𝑟 − 1) = (𝑎 (−1 + 𝑟^(𝑘 + 1) ))/(𝑟 − 1) = 𝑎(𝑟^(𝑘 + 1) − 1)/(𝑟 − 1) Thus, a + ar + ar2 + ……..+ ark – 1 + ark = 𝑎(𝑟^(𝑘 + 1) − 1)/(𝑟 − 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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.