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Question

# Given a geometric progressiona,a1,a2,a3,.........and an arithmetic progressionb,b1,b2,b3,.........with positive terms. The common difference of A.P. and common ratio of G.P. are both positive. Show that there always exists a system of logarithms for which logan−bn=loga−b (for any n)Find base β of the system

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Solution

## Let r be the common ration of G.P. and d the common difference of A.P.Then an=arn ....(1)and bn=b+nd .....(2)Taking logarithms of both sides of (1) to base β(β≠1,β>0), we getlogβan=logβa+nlogβrlogβan−bn=logba+nlogβr−b−nd. by (2).....(3)Now in order that right hand side of (3) reduces tologβa−b, we must have nlogβr−nd=0or logβr=dorr=βd that is ,β=r1/d Hence there exists a system of logarithms to base r1/dsuch that logan−bn=logna−b

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