Mathematics - Sequence & Series Question with Solution | TestHub

${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4\to \text{ G.P. }$
Let a, ar, ${\mathrm{ar}}^2,{\mathrm{ar}}^3\to$ G.P.
Now $a-2,ar-7,a^2-9,{\mathrm{ar}}^3-5\to$ A.P.
$2(\mathrm{ar}-7)=\mathrm{a}-2+{\mathrm{ar}}^2-9$....(i)
$2\left({\mathrm{ar}}^2-9\right)=\mathrm{ar}-7+{\mathrm{ar}}^3-5$....(ii)
$\begin{array}{rl}& \text{ Solving }\mathrm{r}=2,\mathrm{a}=-3\\ & \therefore \text{ Product }={\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4={\mathrm{a}}^4{\mathrm{r}}^6=81\times 64\end{array}$
$\begin{array}{rl}& \text{ Solving }\mathrm{r}=2,\mathrm{a}=-3\\ & \therefore \text{ Product }={\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4={\mathrm{a}}^4{\mathrm{r}}^6=81\times 64\end{array}$