Evaluate ad-bc if a,b,c,d are the terms of a geometric series?

We'll apply the mean theorem of a geometric
series:

b^2 = a*c

sqrt b^2 =
sqrt a*c

b = sqrt a*c (1)

c^2
= b*d

c = sqrt b*d (2)

We'll
multiply bc = sqrt a*b*c*d

But b = a*r, where r is the
common ratio.

c = a*r^2

d =
a*r^3

a*b*c*d =
a*a*r*a*r^2*a*r^3

a*b*c*d =
a^4*r^6

sqrt a*b*c*d = sqrt
a^4*r^6

sqrt a*b*c*d =
a^2*r^3

bc = a^2*r^3 (3)

ad =
a*a*r^3 (4)

We'll subtract (4) from
(3):

a^2*r^3 - a^2*r^3 = 0

So,
the result of the difference is:

ad - bc = 0,
if and only if a,b,c,d are the terms of a geometric
series.