If a, b, c are in G.P., prove that :
(a2 – b2), (b2 – c2), (c2 – d2) are in G.P.
a, b, c, d are in G.P.
Therefore,
bc = ad … (1)
b2 = ac … (2)
c2 = bd … (3)
To prove: (a2 – b2), (b2 – c2), (c2 – d2) are in G.P, we need to prove that:
(a2 – b2) (c2 – d2) = (b2 – c2)2 {deduced using GM relation}
∴ RHS = (b2 – c2)2 = b4 + c4 – 2b2c2
= a2c2 + b2d2 – a2d2 – b2c2 {using equation 2 and 3}
= c2(a2 – b2) – d2(a2 – b2)
= (a2 – b2) (c2 – d2) = LHS
∴ (a2 – b2), (b2 – c2), (c2 – d2) are in G.P
Hence proved.