Insert three geometric means between

To find: Three geometric Mean

Given: The numbers $\frac{1}{3}$ and 432

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

Let $G_{1}, G_{2}$ and $G_{3}$ be the three geometric mean

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

$\Rightarrow r=\left(\frac{b}{a}\right)^{\frac{1}{3+1}}$

$\Rightarrow r=\left(\frac{432}{\left(\frac{1}{3}\right)}\right)^{\frac{1}{2+1}}$

$\Rightarrow r=\left(\frac{432 \times 3}{1}\right)^{\frac{1}{3+1}}$

$\Rightarrow r=(1296)^{\frac{1}{4}}$

⇒ r = 6

$G_{1}=a r=\left(\frac{1}{3}\right) \times 6=2$

$G_{2}=a r^{2}=\left(\frac{1}{3}\right) \times 6^{2}=\left(\frac{1}{3}\right) \times 36=12$

$G_{3}=a r^{3}=\left(\frac{1}{3}\right) \times 6^{3}=\left(\frac{1}{3}\right) \times 216=72$

Three geometric mean between $\frac{1}{3}$ and 432 are 2,12 and 72 .