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math

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if log x base a,log y base a and log z base a are three consecutive terms of an AP,show that x,y and z are consecutive terms of a GP

  • math -

    let A = logax ---> x = a^A
    let B = logay ---> y = a^B
    let C = logaz ---> z = a^C

    but we are told that
    logax , loga y , and loga z
    form an arithmetic sequence
    so A , B, and C form an arithmetic sequence
    and B-A = C-B

    do x, y, and z form a geometric sequence ?
    if so , then y/x must equal z/y

    y/x = a^B / a^A = a^(B-A)
    z/y = a^C / a^B = a^(C-B)
    but B-A = C-B. sp

    y/x =z/y and x, y, z form a geometric sequence

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