If xa = xb/2zb/2 = zc, then prove that are in A.P.
Take logs of each expression, using ln(xa) = a ln(x) etc
ln (p*q) = ln(p) + ln(q):
Given,
xa = xb/2zb/2 = zc
Taking log on each term –
⇒ …(1)
The equality of the first and third expressions tells us that
⇒ …(2)
The second expression is equal to
⇒
⇒ {using equation 1}
∴
Divide through out by ln x
∴ a = b/2 + ab/2c
⇒ 2ac = bc + ab
Dividing the equation by abc –
⇒
From this are in GP.