Question

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

Answer 1

Let the required numbers be $\frac{a}{r},a\mathrm{and}ar.$
Product of the G.P. = 729
$$\begin{gathered} \Rightarrow a^3=729 \\ \Rightarrow a=9 \end{gathered}$$
Sum of the products in pairs = 819
$$\begin{gathered} \Rightarrow \frac{a}{r}\times a+a\times ar+ar\times \frac{a}{r}=819 \\ \Rightarrow a^2\left(\frac{1}{r}+r+1\right)=819 \\ \Rightarrow 81\left(\frac{1+r^2+r}{r}\right)=819 \\ \Rightarrow 9\left(r^2+r+1\right)=91r \\ \Rightarrow 9r^2-82r+9=0 \\ \Rightarrow 9r^2-81r-r+9=0 \\ \Rightarrow \left(9r-1\right)\left(r-9\right)=0 \\ \Rightarrow r=\frac{1}{9},9 \\ \mathrm{Hence},\mathrm{putting}\mathrm{the}\mathrm{values}\mathrm{of}\mathit{}a\mathrm{and}\mathit{}r,\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{numbers}\mathrm{to}\mathrm{be}81,9\mathrm{and}1\mathrm{or}1,9\mathrm{and}81. \end{gathered}$$