Question

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

Solution
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∵ap=a+(p−1)d;aq=a+(q−1)d;ar=a+(r−1)d;as=a+(s−1)d;A/Qpth,qth,rth,sthtermareinG.P∴aqap=araq=asarforaqap=araq⇒aqap−1=araq−1⇒aqap=ar−aqaq−apputtingthevalueoftermweget⇒aqap=(a+(r−1)d)−(a+(q−1)d)(a+(q−1)d)−(a+(p−1)d)=q−rp−q⟶(1)similarlyforaraq=asar⇒araq=r−sq−r⟶(2)from(1)and(2)wecometoknow(p−q),(q−r)and(r−s)areinG.P

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