Let Tr be the rth term of a sequence for r = 1,2,3,..... such that  and Tr > 0  N. If sum of first n terms of this sequence is  and maximum value of T5 is  where p & q are coprime numbers then value of (p + q) is-

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We haveTr+2Tr+1=Tr+1TrSuppose we take r=1T3T2=T2T1 i.e.Third TermSecond Term=Second TermFirst Term which is true for Geometric Progression. Hence given sequence is a G.P.Let a be first term and R be common ratiorth term is given byTr=aRr-1T6=aR5T5=aR4Sum of n terms=aRn-1R-1According to questionaRn-1R-1=1-T6T5naRn-1R-1=1-aR5aR4naRn-1R-1=1-RnaRn-1R-1=-Rn-1a=1-RT5=aR4=1-RR4T5=R4-R5Differentiate with respect to RDT5dR=4R3-5R4=5R345-RCritical points are R=0 and R=45 and method of interval we can seethat R=45 is point maxima.T5=1-RR4T5max=1-45454=2563125=pqp+q=256+3125=3381 AnswerIf you have any specific doubt in this solution or answer is not matching,feel free to ask.

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