Insert two geometric means between 9 and 243.

To find: Two geometric Mean

Given: The numbers are 9 and 243

Formula used: (i) $r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$, where $n$ is the number of

geometric mean

Let $G_{1}$ and $G_{2}$ be the three geometric mean

Then $r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

$\Rightarrow r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

$\Rightarrow r=\left(\frac{243}{9}\right)^{\frac{1}{2+1}}$

$\Rightarrow r=27^{\frac{1}{3}}$

⇒ r = 3

$G_{1}=a r=9 \times 3=27$

$G_{2}=a r^{2}=9 \times 3^{2}=9 \times 9=81$

Two geometric mean between 9 and 243 are 27 and 81.