If a, b, c are in G.P; a, x, b are in A.P. and b, y, c are also in A.P. Prove that : .

Given, a, x, b are in A.P. ∴ b - x = x - a ⇒ 2x = a + b ⇒ x = .........(1) Given, b, y, c are in A.P. ∴ c - y = y - b ⇒ 2y = b + c ⇒ y = ...........(2) Given, a, b and c are in G.P. ⇒ ⇒ b 2 = ac ..........(3) To prove : Substituting value of x and y from equation (1) and (2), in L.H.S. of above equation : Hence, proved that .