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Question

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is


A

pqqr

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B

qrpq

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C

pqr

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D

none of these

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Solution

The correct option is B

qrpq


(b) qrpq

Let a be the first term and d be the common difference of the given A.P.

Then, we have :

pth term, ap=a+(p1)d

qth term, aq=a+(q1)d

rth term, ar=a+(r1)d

Now, according to the question the pth, the qth

and the rth terms are in G.P.

(a+(q1)d)2=(a+(p1)d)×(a+(r1)d)

a2+2a(q1)d+((q1)d)2=a2+ad(r1+p1)+(p1)(r1)d2

ad(2qrp+2)+d2(q22q+1pr+p+r1)=0

a(2qrp)+d(q22qpr+p+r)

=0 ( d cannot be 0 )

a=(q22qpr+p+r)d(2qrp)

Common ratio, r=apaq

=a+(q1)da+(p1)d=(q22qpr+p+r)d+(q1)d(q22qpr+p+r)dp+r2q+(p1)d

=q22qpr+p+r+pq+rq2q2pr+2qq22qpr+p+r+p2+prq2pq2pr+2q

=p(qr)q(qr)(pq)2=(pq)(qr)(pq)2=(qr)(pq)


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