If the fourth, seventh and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.

Given, a₄ = x, a₇ = y and a₁₀ = z.

By formula, a_n = ar^{n - 1}.

⇒ x = a₄ = a(r)³

⇒ y = a₇ = a(r)⁶

⇒ z = a₁₀ = a(r)⁹

From above equations,

$\Rightarrow \dfrac{y}{x} = \dfrac{ar^6}{ar^3} = r^3. \\[1em] \Rightarrow \dfrac{z}{y} = \dfrac{ar^9}{ar^6} = r^3.$

The above equations prove that the common ratio between x, y and z is r³.

Hence, x, y and z are in G.P. with common ratio = r³.