Answer :

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

Cross multiplying (2) and expanding, (b + cx)(c – dx) = (c + dx)(b – cx)
So, from (i) b, is the geometric mean of a and b. So, a, b, c are in GP.

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.