It is given that,

On cross multiplying, we get -

(a + bx)(b - cx) = (b + cx)(a - bx)

⇒ ab - acx + b²x - bcx² = ab - b²x + acx - bcx²

⇒ 2b²x = 2acx

⇒ b² = ac

…(1)

Also,

On cross multiplying, we get -

⇒ (b + cx)(c - dx) = (c + dx)(b - cx)

⇒ bc - bdx + c²x - cdx² = bc + bdx - c²x - cdx²

⇒ 2c²x = 2bdx

⇒ c² = bd

…(2)

From (1) and (2), we obtain

Thus, a, b, c, and d are in G.P.