Posted by **jessica** on Saturday, February 16, 2013 at 7:29am.

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 -
**Reiny**, Saturday, February 16, 2013 at 8:03am
let A = log_{a}x ---> x = a^A

let B = log_{a}y ---> y = a^B

let C = log_{a}z ---> z = a^C

but we are told that

log_{a}x , log_{a} y , and log_{a} 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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