Question

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

Answer 1

let A = log_ax ---> x = a^A
let B = log_ay ---> y = a^B
let C = log_az ---> z = a^C

but we are told that
log_ax , 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