Question 8
If pth, qth, rth and sth terms of an A.P. are in G.P, then show that (p – q), (q – r), (r – s) are also in G.P.
We know that the nth term of AP is a + (n – 1)d
i.e. an = a + (n – 1)d
It is given that ap, aq, ar & as in GP
i.e. their common ratio is same
So,𝑎_𝑞/𝑎_𝑝 = (ar )/qq = 𝑎𝑠/𝑎𝑟
We need to show (p – q), (q – r), (r – s) are in GP
i.e. we need to show their common ratio is same
i.e. we need to show (q − r )/(p − q) = (r − s )/(q − r)
It is given
(aq )/qp = (ar )/qq = (as )/ar
From (A) & (B)
(𝑞 − 𝑟)/(𝑝 − 𝑞) = (𝑟 − 𝑠)/(𝑞 − 𝑟)
Hence (p – q), (q – r), (r – s) are in GP
Hence proved
Made by
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
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