Question

If a, b, c are in G.P., prove that the following are also in G.P.:
(i) a², b², c²
(ii) a³, b³, c³
(iii) a² + b², ab + bc, b² + c²

Answer 1

a, b and c are in G.P.
∴ $b^2=ac.......\left(1\right)$
$$\begin{gathered} \left(\mathrm{i}\right){\left(b^2\right)}^2={\left(ac\right)}^2\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow {\left(b^2\right)}^2=a^2{c}^2 \\ \mathrm{Therefore},a^2,b^2\mathrm{and}{c}^2\mathrm{are}\mathrm{also}\mathrm{in}\mathrm{G}.\mathrm{P}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right){\left(b^3\right)}^2={\left(b^2\right)}^3={\left(ac\right)}^3\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow {\left(b^3\right)}^2=a^3{c}^3 \\ \mathrm{Therefore},a^3,b^3\mathrm{and}{c}^3\mathrm{are}\mathrm{also}\mathrm{in}\mathrm{G}.\mathrm{P}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right){\left(ab+bc\right)}^2={\left(ab\right)}^2+2a{b}^{2}c+{\left(bc\right)}^2 \\ \Rightarrow {\left(ab+bc\right)}^2={\left(ab\right)}^2+a{b}^{2}c+a{b}^{2}c+{\left(bc\right)}^2 \\ \Rightarrow {\left(ab+bc\right)}^2=a^2{b}^2+ac\left(ac\right)+b^2\left(b^2\right)+b^2{c}^2\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow {\left(ab+bc\right)}^2=a^2\left(b^2+c^2\right)+b^2\left(b^2+c^2\right) \\ \Rightarrow {\left(ab+bc\right)}^2=\left(b^2+c^2\right)\left(a^2+b^2\right) \\ \mathrm{Therefore},\left(a^2+b^2\right),\left(b^2+c^2\right)\mathrm{and}\left(ab+bc\right)\mathrm{are}\mathrm{also}\mathrm{in}\mathrm{G}.\mathrm{P}. \end{gathered}$$