Insert four geometric means between 6 and 192.

Solution:

To find: Four geometric Mean

Given: The numbers 6 and 192

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

geometric mean

Let $G_{1}, G_{2}, G_{3}$ and $G_{4}$ be the three geometric mean

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

$\Rightarrow r=\left(\frac{b}{a}\right)^{\frac{1}{4+1}}$

$\Rightarrow r=\left(\frac{192}{6}\right)^{\frac{1}{4+1}}$

$\Rightarrow r=(32)^{\frac{1}{5}}$

⇒ r = 2

$G_{1}=a r=6 \times 2=12$

$G_{2}=a r^{2}=6 \times 2^{2}=24$

$G_{3}=a r^{3}=6 \times 2^{3}=48$

$G_{4}=a r^{4}=6 \times 2^{4}=96$

Four geometric mean between 6 and 192 are 12, 24, 48 and 96.