# Geometric seriesCalculateĀ ad - bc if a,b,c,d are the terms of a geometric series.

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We need to find the value of ad - bc if a,b,c,d are the terms of a geometric series.

As a, b, c and d are terms of a geometric series. b=a*r, c=a*r^2 and d = a*r^3

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

We see that a*d = b*c, a*d - b*c = 0

**If a,b,c,d are the terms of a geometric series, 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.**