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Given-that-a-b-c-are-3-consecutive-term-of-a-Geometric-sequence-f-n-show-that-log-a-logb-logc-are-the-first-3-terms-of-an-Arithmetic-SequenceP-n-




Question Number 38154 by Rio Mike last updated on 22/Jun/18
Given that     a,b,c are 3 consecutive term of   a Geometric sequence f(n) , show  that log a,logb,logc are the first   3 terms of an Arithmetic SequenceP(n).
Giventhata,b,care3consecutivetermofaGeometricsequencef(n),showthatloga,logb,logcarethefirst3termsofanArithmeticSequenceP(n).
Answered by Rasheed.Sindhi last updated on 22/Jun/18
a,b=ar,c=ar^2   log a ,log b=log ar=log a +log r  log c=log (ar^2 )=log a+2log r  (i)log b −log a=log a +log r−log a=log r  (ii)log c−log b=(log a+2log r)−(log a +log r)                =log r  From (i) & (ii)      log b −log a=log c−log b=log r  ∴ log a , log b & log c are in AP
a,b=ar,c=ar2loga,logb=logar=loga+logrlogc=log(ar2)=loga+2logr(i)logbloga=loga+logrloga=logr(ii)logclogb=(loga+2logr)(loga+logr)=logrFrom(i)&(ii)logbloga=logclogb=logrloga,logb&logcareinAP

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