We know that all the consecutive terms of a geometric progression have a common ratio. And it is possible to write consecutive terms as the previous term multiplied by the common ratio.
If a, b, c and d are terms of a geometric series, they can be expressed as :
b = ar
c = ar^2
and d= ar^3, where r is the common ratio between the terms.
So a*d - b*c = a*ar^3 - ar*ar^2 = a^2*r^3 - a^2*r^3 = 0
Therefore ad - bc is equal to zero.
We assume that the a,b,c and d are the cosecutive terms of a geometric progression.
So aa =a
a2 = ar = b
a3 = ar^2 = c
a4 = ar^3 = d.
Therefore ad-bc = a*ar^3 - ar*ar^2 = a^2r^3-a^2*r^3 = 0.
Threfore ad-bc = 0.
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.