Question

If A₁, A₂ be two AM's and G₁, G₂ be two GM's between a and b, then find the value of $\frac{A_1+A_2}{G_1{G}_2}$.

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}{A}_1\mathrm{and}{A}_2\mathrm{are}\mathrm{the}\mathrm{A}.\mathrm{M}.\mathrm{s}\mathrm{between}\mathit{}a\mathrm{and}b. \\ \mathrm{Thus},a\mathit{}\mathit{,}\mathit{}{A}_{\mathit{1}}\mathit{,}\mathit{}{A}_{\mathit{2}}\mathrm{and}\mathit{}b\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{common}\mathrm{difference}\mathrm{d}. \\ \mathrm{Here},d=\frac{b\mathit{-}a}{\mathit{3}} \\ \therefore A_1=a\mathit{}\mathit{+}\mathit{}\frac{b\mathit{-}a}{\mathit{3}}\mathit{}\mathit{=}\frac{2a+b}{3} \\ \mathrm{and}{A}_2=a\mathit{}\mathit{+}\mathit{}\frac{2\left(b-a\right)}{3}=\frac{a+2b}{3} \\ \mathrm{It}\mathrm{is}\mathrm{also}\mathrm{given}\mathrm{that}{\mathrm{G}}_1\mathrm{and}{\mathrm{G}}_2\mathrm{are}\mathrm{the}\mathrm{G}.\mathrm{M}.\mathrm{s}\mathrm{between}a\mathrm{and}b. \\ \mathrm{Thus},\mathit{}a\mathit{}\mathit{,}\mathit{}{G}_{\mathit{1}}\mathit{,}\mathit{}{G}_{\mathit{2}}\mathrm{and}\mathit{}b\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}.\mathrm{with}\mathrm{common}\mathrm{ratio}\mathit{}r. \\ \mathrm{Here},r\mathit{}={\left(\frac{b}{a}\right)}^{\frac{1}{3}} \\ \therefore G_{\mathit{1}}\mathit{}\mathit{=}\mathit{}a{\left(\frac{b}{a}\right)}^{\frac{1}{3}}\mathit{}\mathit{=}\mathit{}{b}^{\frac{\mathit{1}}{\mathit{3}}}\mathit{}{a}^{\frac{1}{3}}\mathit{} \\ \mathrm{and}{G}_{\mathit{2}}\mathit{}\mathit{=}\mathit{}a{\left[{\left(\frac{b}{a}\right)}^{\frac{1}{3}}\mathit{}\right]}^{\mathit{2}}\mathit{=}\mathit{}{b}^{\frac{1}{3}}\mathit{}{a}^{\frac{1}{3}} \\ \Rightarrow \frac{A_{\mathit{1}}\mathit{}+\mathit{}{A}_{\mathit{2}}}{G_{\mathit{1}}\mathit{}{G}_{\mathit{2}}}\mathit{}\mathit{=}\mathit{}\frac{\frac{2a\mathit{+}b}{3}+\frac{a\mathit{+}\mathit{2}b}{3}}{\mathit{}{b}^{\frac{1}{3}}\mathit{}{a}^{\frac{1}{3}}\mathit{}\times \mathit{}{b}^{\frac{1}{3}}\mathit{}{a}^{\frac{1}{3}}\mathit{}}\mathit{}\mathit{=}\mathit{}\frac{a+b}{ab} \end{gathered}$$