Question

Find the geometric means of the following pairs of numbers:
(i) 2 and 8
(ii) a³b and ab³
(iii) −8 and −2

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Let}\mathrm{the}\mathrm{G}.\mathrm{M}.\mathrm{between}2\mathrm{and}8\mathrm{be}G. \\ \mathrm{Then},2,G\mathrm{and}8\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore G^2=2\times 8 \\ \Rightarrow G^2=16 \\ \Rightarrow G=\pm \sqrt{16} \\ \Rightarrow G=\pm 4 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}\mathrm{the}\mathrm{G}.\mathrm{M}.\mathrm{between}{a}^{3}b\mathrm{and}a{b}^3\mathrm{be}G. \\ \mathrm{Then},a^{3}b,G\mathrm{and}a{b}^3\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore G^2=a^{3}b\times a{b}^3 \\ \Rightarrow G^2=a^4{b}^4 \\ \Rightarrow G=\sqrt{a^4{b}^4} \\ \Rightarrow G=a^2{b}^2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Let}\mathrm{the}\mathrm{G}.\mathrm{M}.\mathrm{between}-8\mathrm{and}-2\mathrm{be}G. \\ \mathrm{Then},-8,G\mathrm{and}-2\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore G^2=\left(-8\right)\left(-2\right) \\ \Rightarrow G^2=16 \\ \Rightarrow G=\pm \sqrt{16} \\ \Rightarrow G=4,-4 \end{gathered}$$