If akak-1 + ak-1​ak-2​ = 2a​k ak-2  , k>=3 and a1=1 , here Sp= sigma (k = 1 to k = p) 1/aand given that S2p/Sdoes not depend on p then 1/a2016 is = ?

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We have:akak-1+ak-1 ak-2=2ak ak-2Put k=3a3a2+a2 a1=2a3 a1a3a2+a2=2a3 As a1=1 a3=a22-a2 ;iSimilarly after putting k=4, we get:a4a3+a3 a2=2a4 a2a4=a3 a22a2-a3a4=a22-a2 a22a2-a22-a2=a223a2-2a22=a23-2a2 ;iiAlso according to ques:S2pSp is independent of pS2S1=S4S21+1a21=1+1a2+2-a2a2+3-2a2a21+1a21+1a22=1+1a2+2-a2a2+3-2a2a21+1a22-1+1a2=2-a2+3-2a2a21+1a21+1a2-1=5-3a2a21+1a21a2=5-3a2a2a2+1a2=5-3a23a22-4a2+1=03a22-3a2-a2+1=03a2a2-1-1a2-1=0a2-13a2-1=0a2=1 or a2=13But if a2=1, then series will become 1,1,1,1,1,1,1,Hence we take a2=13Putting value of a2 in i and ii, we get:a3=132-13=15a4=133-23=17Hence our series is 1,13,15,17,.....As denominator's term are in A.P. wil first term=1 and common difference=2Hence we get: a2016=11+2016-12=14031Hence1 a2016=4031 AnswerNote: In case series is taken as 1,1,1,1,1,1,..., then 1 a2016=1

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