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If a, b, c, d are in a geometric sequence, then show that (a – b + c) (b + c + d) = ab + bc + cd.

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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.

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