If pth, qth and rth terms of an A.P. are in G.P. whose common ratio is k, then the root of equation
q - r x 2 + r - p x + p - q = 0 other than unity is :
(1) k 2               (2) k                 (3) 2k                (4) None of these

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Please find below the solution to the asked query:

pth, qth and rth of A.P. are in G.P.If a is first term and d is common difference, thenTp=a+p-1dTq=a+q-1dTr=a+r-1dAs they are in G.P.TqTp=TrTqa+q-1da+p-1d=a+r-1da+q-1d=k Common ratioa+q-1da+p-1d-1=a+r-1da+q-1d-1a+q-1d-a-p-1da+p-1d=a+r-1d-a-q-1da+q-1ddq-pa+p-1d=dr-qa+q-1da+q-1da+p-1d=r-qq-p=q-rp-qk=q-rp-qNowq-rx2+r-px+p-q=0If we put x=1 in L.H.S> it becomes q-r+r-p+p-q=0 , hence1 is root of above equation, let other root be αProduct of roots=1× α=p-qq-rα=p-qq-rα=1k   As k=q-rp-qOptiond

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