JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Sequences and Series

Write the geometric progression as $a,ar,a{r}^2,a{r}^3$. Subtracting $2,7,9,5$ yields an arithmetic progression, so its successive differences are equal: from the first equality $a(r-1{)}^2=-3$, and from the second $ar(r-1{)}^2=-6$. Dividing one by the other gives $r=2$, and then $a=-3$. The four numbers are $-3,-6,-12,-24$, with product $5184$, so $\frac{1}{24}(x_1{x}_2{x}_3{x}_4)=\frac{5184}{24}=216$.