(Given data in the question) → (1)

Cross multiplying (1) and expanding,

(a + bx)(b – cx) = (b + cx)(a-bx)

ab – acx + b²x – bcx² = ba –b²x + acx – bcx²

2b²x = 2acx

b² = ac → (i)

If three terms are in GP, then the middle term is the Geometric Mean of first term and last term.

→ b² = ac

So, from (i) b, is the geometric mean of a and b.

So, a, b, c are in GP.

(Given data in the question) → (2)

Cross multiplying (2) and expanding,

(b + cx)(c – dx) = (c + dx)(b – cx)

bc – bdx + c²x – cdx² = cb – c²x + bdx – dcx²

2c²x = 2bdx

c² = bd → (ii)

So, from (ii), c is the geometric mean of b and d.

So, b, c, d is in GP.

∴a, b, c, d are in GP.