Question

Find the geometric means of the following pairs of numbers.

(i) 2 and 8

(ii) $a^{3}b$ and $a{b}^3$

(iii) - 8 and - 2

Answer 1

(i) 2 and 8

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) $a^{3}b$ and $a{b}^3$

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) - 8 and - 2

Let the G.M. between - 8 and - 2 be G.

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

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

$\Rightarrow G^2=16$

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

$\Rightarrow G=4,-4$