If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is

It is given that x, 2y, 3z are in A.P

∴ 2y – x = 3z – 2y

⇒ 2y + 2y = x + 3z

⇒ 4y = x + 3z

⇒ x = 4y – 3z …(i)

and it is also given that x, y, z are in G.P

∴ Common ratio …(ii)

∴ y × y = x × z

⇒ y² = xz …(iii)

Putting the value of x = 4y – 3z in eq. (iii), we get

y² = (4y – 3z)(z)

⇒ y² = 4yz – 3z²

⇒ 3z² – 4yz + y² = 0

⇒ 3z² – 3yz – yz + y² = 0

⇒ 3z(z – y) – y(z – y) = 0

⇒ (3z – y)(z – y) = 0

⇒ 3z – y = 0 & z – y = 0

⇒ 3z = y & z = y but z and y are distinct numbers

& z ≠ y

[from eq. (ii)]

Hence, the correct option is (b)