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Question -

If the pth, qth┬аand rth┬аtermsof a G.P. are┬аa, b┬аand┬аc, respectively. Prove thataq-r┬аbr-p┬аcp-q┬а= 1



Answer -

LetтАЩs take┬аA┬аtobe the first term and┬аR┬аto be the common ratio of the G.P.

Then according to thequestion, we have

ARpтАУ1┬а=┬аa

ARqтАУ1┬а=┬аb

ARrтАУ1┬а=┬аc

Then,

aqтАУr┬аbrтАУp┬аcpтАУq

=┬аAqтАУr┬а├Ч┬аR(pтАУ1)(qтАУr)┬а├Ч ArтАУp┬а├Ч┬аR(qтАУ1)(rтАУp)┬а├Ч┬аApтАУq┬а├Ч┬аR(r┬атАУ1)(pтАУq)

=┬аAq┬атАУ┬аr┬а+┬аr┬атАУ┬аp┬а+┬аp┬атАУ┬аq┬а├Ч┬аR┬а(pr┬атАУ┬аpr┬атАУ┬аq┬а+┬аr)+ (rq┬атАУ┬аr┬а+┬аp┬атАУ┬аpq) +(pr┬атАУ┬аp┬атАУ┬аqr┬а+┬аq)

=┬аA0┬а├Ч┬аR0

= 1

Hence proved.

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