Let x_1, x_2, x_3, x_4 be in a geometric progression. 2,7,9,5 are subtracted respectively from x_1, x_2, x_3 x_4 then the resulting numbers are in an…

$$\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4 \rightarrow \text { G.P. }$$x1​,x2​,x3​,x4​→ G.P. 

Let a, ar, $\mathrm{ar}^2, \mathrm{ar}^3 \rightarrow$ar2,ar3→ G.P.

Now $a-2, a r-7, a^2-9, \mathrm{ar}^3-5 \rightarrow$a−2,ar−7,a2−9,ar3−5→ A.P.

$2(\mathrm{ar}-7)=\mathrm{a}-2+\mathrm{ar}^2-9$2(ar−7)=a−2+ar2−9....(i)

$2\left(\operatorname{ar}^2-9\right)=\mathrm{ar}-7+\mathrm{ar}^3-5$2(ar2−9)=ar−7+ar3−5....(ii)

$$\begin{aligned} & \text { Solving } \mathrm{r}=2, \mathrm{a}=-3 \\ & \therefore \text { Product }=\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4=\mathrm{a}^4 \mathrm{r}^6=81 \times 64\end{aligned}$$​ Solving r=2,a=−3∴ Product =x1​,x2​,x3​,x4​=a4r6=81×64​

$$\begin{aligned} & \text { Solving } \mathrm{r}=2, \mathrm{a}=-3 \\ & \therefore \text { Product }=\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4=\mathrm{a}^4 \mathrm{r}^6=81 \times 64\end{aligned}$$​ Solving r=2,a=−3∴ Product =x1​,x2​,x3​,x4​=a4r6=81×64​