Find the geometric means of the following pairs of numbers:

Solution:

(i) Let the G.M. between 2 and 8 be $G$.

Then, $2, G$ and 8 are in G.P.

$\therefore G^{2}=2 \times 8$

$\Rightarrow G^{2}=16$

$\Rightarrow G=\pm \sqrt{16}$

$\Rightarrow G=\pm 4$

(ii) Let the G.M. between $a^{3} b$ and $a b^{3}$ be $G .$

Then, $a^{3} b, G$ and $a b^{3}$ are in G.P.

$\therefore G^{2}=a^{3} b \times a b^{3}$

$\Rightarrow G^{2}=a^{4} b^{4}$

$\Rightarrow G=\sqrt{a^{4} b^{4}}$

$\Rightarrow G=a^{2} b^{2}$

(iii) Let the G.M. between $-8$ and $-2$ be $G$.

Then, $-8, G$ and $-2$ are in G.P.

$\therefore G^{2}=(-8)(-2)$

$\Rightarrow G^{2}=16$

$\Rightarrow G=\pm \sqrt{16}$

$\Rightarrow G=4,-4$