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# If pth,qth,rth and sth terms of an A.P are in G.P, then show that (p−q)(q−r)(r−s) are also in G.P.

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#### Given,ap=a+(p−1)daq=a+(q−1)dar=a+(r−1)das=a+(s−1)dap,aq,ar,as are in G.P⇒aqap=asarconsider,aqap=araqsubtracting 1 on both sides, we get,aqap−1=araq−1aq−apar−aq=apaqa+(q−1)d−[a+(p−1)d]a+(r−1)d−[a+(q−1)d]=apaqq−rp−q=apaq............(1)now consider,araq=asarsubtracting 1 on both sides, we get,araq−1=asar−1ar−asaq−ar=araqa+(r−1)d−[a+(s−1)d]a+(q−1)d−[a+(r−1)d]=araqr−sq−r=araqFrom (1)r−sq−r=aqap............(2)From (1) and (2), we have,apaq=aqapTherefore (p−q),(q−r),(r−s) are in G.P

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