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Question

Let a1,a2,a3......... are in A.P such that ap, aq,ar are in both A.P and GP then aq:ap is equal to


A

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B

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C

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D

None of these

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Solution

The correct option is C


Let the common ration of the G.P be α and ap = x

Then,

aq = xα

ar = xα2

We want to find aq:ap or aqap

aqap = xαx = α

We have to find α in terms of p,q and r,because the options are in terms of them.

Sinceap,aq,ar in A.P,we can express them in terms of p,q and respectively.

ap = a + (p-1)d

aq = a + (q - 1)d

ar = a + (r - 1)d

It also has the common difference and the first term.

Now,if we can establish some relation between the α and ap,aq,ar without a and d,our job will be done.For that,

consider αrαqαqαp

= xα2xαxαx

= α(xαx)xαx

= α = aq:ap

So, aq:ap = αrαqαqαp

= α(r1)d(α+(q1)d)α+(q1)d(α+(p1)d)

= (rq)d(qp)d

=rqqp --------- C


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