Question 12 - End-of-Chapter Exercises - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Question 12 If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P. For a GP, We know that nth term = an = arn – 1 Let’s find the 4th, 10th and 16th terms 4th term is x 4th term = a4 x = ar4 – 1 x = ar3 10th term is y 10th term = a10 y = ar10 – 1 y = ar9 16th term is z 16th term = a16 z = ar16 – 1 z = ar15 We need to show x, y, z are in GP If x, y, z are in GP, their common ratio is same So, we need to show (2^𝑛𝑑 𝑡𝑒𝑟𝑚)/(1^𝑠𝑡 𝑡𝑒𝑟𝑚)=" " (3^𝑟𝑑 𝑡𝑒𝑟𝑚)/(2^𝑛𝑑 𝑡𝑒𝑟𝑚) 𝒚/𝒙 = 𝒛/𝒚 Calculating 𝒚/𝒙 𝒚/𝒙 Putting y = ar9 & x = ar3 = (𝑎𝑟^9)/(𝑎𝑟^3 ) = 𝑟^(9−3) = 𝒓^𝟔 Calculating 𝒛/𝒚 𝒛/𝒚 Putting z = ar15 & y = ar9 = (𝑎𝑟^15)/(𝑎𝑟^9 ) = 𝑟^(15 − 9) = 𝒓^𝟔 Thus, 𝒚/𝒙 = 𝒛/𝒚 Since common ratio is same Therefore, x, y, z are in G.P Hence proved