Given, a, b, c, d are in a geometric sequence.
Let a = a, b = ar, c = ar2, d = ar3
To prove, (a – b + c) (b + c + d) = ab + bc + cd
L.H.S. = (a – b + c)(b + c + d)
= (a – ar + ar2) (ar + ar2 + ar3)
= a (1 – r + r2)ar (1 + r + r2)
= a2 r (1 – r + r2) (1 + r + r2)
= a2 r (1 + r2 + r4)
= a2r + a2r3 + a2r5
= a (ar) + ar (ar2) + ar2 (ar3)
= ab + bc + cd
= R.H.S.
L.H.S. = R.H.S., Hence proved.