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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-

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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)logb−loga=loga+logr−loga=logr(ii)logc−logb=(loga+2logr)−(loga+logr)=logrFrom(i)&(ii)logb−loga=logc−logb=logr∴loga,logb&logcareinAP

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