If a, b and c are in A.P. a, x, b are in G.P. whereas b, y and c are also in G.P.

Show that : x², b², y² are in A.P.

Given,

a, b, c are in A.P.

⇒ 2b = a + c ……..(i)

a, x, b are in G.P.

⇒ x² = ab ……..(ii)

b, y and c are in G.P.

⇒ y² = bc ………(iii)

Adding (ii) and (iii) we get,

⇒ x² + y² = ab + bc

⇒ x² + y² = b(a + c)

⇒ x² + y² = b.2b (From i)

⇒ x² + y² = 2b².

Hence, proved that x², b², y² are in A.P.