If x, y and z are in G.P., show that x⁴, y⁴ and z⁴ are also in G.P.

Given, x, y and z are in G.P.

$\therefore \dfrac{y}{x} = \dfrac{z}{y}$

∴ y² = xz = m (let) …..(1)

New terms = x⁴, y⁴ and z⁴.

For the terms to be in G.P., common ratio between terms must be equal,

$\Rightarrow \dfrac{y^4}{x^4} = \dfrac{z^4}{y^4} \\[1em] \Rightarrow (y^4)^2 = x^4.z^4 \\[1em] \Rightarrow (y^2)^4 = (xz)^4$

Substituting value from equation (1), we get :

⇒ m⁴ = m⁴.

Since, L.H.S. = R.H.S.

Hence, proved that x⁴, y⁴ and z⁴ are in G.P.