Question

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

Answer 1

Let the first term of a G.P be a and its common ratio be r.
$$\begin{gathered} \therefore a_1+a_2+a_3=56 \\ \Rightarrow a+ar+a{r}^2=56 \\ \Rightarrow a\left(1+r+r^2\right)=56 \\ \Rightarrow a=\frac{56}{1+r+r^2}.......\left(\mathrm{i}\right) \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{the}\mathrm{question}: \\ \mathrm{a}-1,\mathrm{ar}-7\mathrm{and}{\mathrm{ar}}^2-21\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}. \\ \therefore 2\left(\mathrm{ar}-7\right)=\mathrm{a}-1+{\mathrm{ar}}^2-21 \\ \Rightarrow 2\mathrm{ar}-14={\mathrm{ar}}^2+\mathrm{a}-22 \\ \Rightarrow {\mathrm{ar}}^2-2\mathrm{ar}+\mathrm{a}-8=0 \\ \Rightarrow \mathrm{a}{\left(1-\mathrm{r}\right)}^2=8 \\ \Rightarrow \mathrm{a}=\frac{8}{{\left(1-\mathrm{r}\right)}^2}.......\left(\mathrm{ii}\right) \\ \mathrm{Equating}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right): \\ \Rightarrow \frac{8}{{\left(1-\mathrm{r}\right)}^2}=\frac{56}{1+\mathrm{r}+{\mathrm{r}}^2} \\ \Rightarrow 8\left(1+\mathrm{r}+{\mathrm{r}}^2\right)=56\left(1+{\mathrm{r}}^2-2\mathrm{r}\right)\Rightarrow 1+\mathrm{r}+{\mathrm{r}}^2=7\left(1+{\mathrm{r}}^2-2\mathrm{r}\right) \\ \Rightarrow 1+\mathrm{r}+{\mathrm{r}}^2=7+7{\mathrm{r}}^2-14\mathrm{r} \\ \Rightarrow 6{\mathrm{r}}^2-15\mathrm{r}+6=0 \\ \Rightarrow 3\left(2{\mathrm{r}}^2-5\mathrm{r}+2\right)=0 \\ \Rightarrow 2{\mathrm{r}}^2-4\mathrm{r}-\mathrm{r}+2=0 \\ \Rightarrow 2\mathrm{r}(\mathrm{r}-2)-1(\mathrm{r}-2)=0 \\ \Rightarrow (\mathrm{r}-2)(2\mathrm{r}-1)=0 \\ \Rightarrow \mathrm{r}=2,\frac{1}{2} \\ \mathrm{When}\mathrm{r}=2,\mathrm{a}=8.[\mathrm{Using}(\mathrm{ii}\left)\right] \\ \mathrm{And},\mathrm{the}\mathrm{required}\mathrm{numbers}\mathrm{are}8,16\mathrm{and}32. \\ \mathrm{When}\mathrm{r}=\frac{1}{2},\mathrm{a}=32.[\mathrm{Using}(\mathrm{ii}\left)\right] \\ \mathrm{And},\mathrm{the}\mathrm{required}\mathrm{numbers}\mathrm{are}32,16\mathrm{and}8. \end{gathered}$$