Question

If a, b, c and d are in G.P. show that (a 2 + b 2 + c 2 ) (b 2 + c 2 + d 2 ) = (ab + bc + cd)2.

Answer 1

a, b, c, d are in G.P.
Therefore,
bc = ad … (1)
b 2 = ac … (2)
c 2 = bd … (3)
It has to be proved that,
(a 2 + b 2 + c 2 ) (b 2 + c 2 + d 2 ) = (ab + bc-cd) 2
R.H.S.
= (ab + bc + cd) 2
= (ab + ad + cd) 2 [Using (1)]
= [ab + d (a + c)]2
= a 2b 2 + 2abd (a + c) + d 2 (a + c) 2
= a 2b 2 +2a 2bd + 2acbd + d 2 (a 2 + 2ac + c 2 )
= a 2b 2 + 2a 2 c 2 + 2b 2 c 2 + d 2a 2 + 2d 2b 2 + d 2 c 2 [Using (1) and (2)]
= a 2b 2 + a 2 c 2 + a 2 c 2 + b 2 c 2 + b 2 c 2 + d 2a2 + d 2b 2 + d 2b 2 + d 2 c 2
= a 2b 2 + a 2 c 2 + a 2d 2 + b 2 × b 2 + b 2 c 2 + b 2d 2 + c2b 2 + c 2 × c 2 + c 2d2
[Using (2) and (3) and rearranging terms]
= a 2 (b 2 + c 2 + d 2 ) + b 2 (b 2 + c 2 + d 2 ) + c 2 (b 2+ c 2 + d 2 )
= (a 2 + b 2 + c 2 ) (b 2 + c 2 + d 2 )
= L.H.S.
∴ L.H.S. = R.H.S.
∴ (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)2