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Question

If a,b,c are the pth,qth and rth terms of a GP, show that (qr)loga+(rp)logb+(pq)logc=0.

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Solution

Let A be the first term and R be the common ratio of G.P
A/Q, a=ARp1b=ARq1c=ARr1
now, loga=logARp1loga=logA+(p1)logR
similarly, logb=logA+(q1)logRlogc=logA+(r1)logR
Putting the above value in the equation, we get;
LHS=(qr)loga+(rp)logb+(pq)logc=(qr)[logA+(p1)logR]+(rp)[logA+(q1)logR]+(pq)[logA+(r1)logR]=logA[(qr)+(rp)+(pq)]+logR[(qr)(p1)+(rp)(q1)+(pq)(r1)]=0.logA+0.logR=0=RHS


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