Question

If pth,qth,rth and sth terms of an A.P are in G.P, then show that (pq)(qr)(rs) are also in G.P.

Solution
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Given,

ap=a+(p1)d

aq=a+(q1)d

ar=a+(r1)d

as=a+(s1)d

ap,aq,ar,as are in G.P

aqap=asar

consider,

aqap=araq

subtracting 1 on both sides, we get,

aqap1=araq1

aqaparaq=apaq

a+(q1)d[a+(p1)d]a+(r1)d[a+(q1)d]=apaq

qrpq=apaq............(1)

now consider,

araq=asar

subtracting 1 on both sides, we get,

araq1=asar1

arasaqar=araq

a+(r1)d[a+(s1)d]a+(q1)d[a+(r1)d]=araq

rsqr=araq

From (1)

rsqr=aqap............(2)

From (1) and (2), we have,

apaq=aqap

Therefore (pq),(qr),(rs) are in G.P

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