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Asked by vk1994 | 22nd Aug, 2010, 11:07: AM

Expert Answer:

Let an be the nth term of an AP.
ap = a + (p-1)d    ....(1)
aq = a + (q-1)d    ....(2)
ar = a + (r-1)d      ....(3)
as = a + (s-1)d      ....(3)
We can do (1) - (2) to get (p-q), and similarly can obtain expression for other terms.
(p-q) = (ap -  aq)/d = ap(1 - R)/d       .... since pth, qth, rth and sth terms are also in GP and R is the common ratio.
(q-r) = (aq -  ar )/d = aq (1 -  R )/d
(r-s) = (ar -  as )/d = ar(1 -  R )/d
Now,
(p-q)/(q-r) = ap/a = R
(q-r)/(r-s) = aq /ar = R.
Hence proved.
Regards,
Team,
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Answered by  | 22nd Aug, 2010, 09:54: PM

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