Correct option is: (4) 216
\( x_1=a ; x_2=a r ; x_3=a r^2 ; x_4=a r^3 \)
\(a-2, a r-7, a r^2-9, a r^3-5 \rightarrow \text { A.P. } \)
\( a_2-a_1=a_3-a_2 \)
\( (a r-7)-(a-2)=\left(a r^2-9\right)-(a r-7) \)
\(=a(r-1)-5=a r(r-1)-2 \)
\( a(r-1)(r-1)=-3 \quad \cdots(i)\)
\( a_2-a_1=a_4-a_3 \)
\( (a r-7)-(a-2)=\left(a r^3-5\right)-\left(a r^2-9\right) \)
\( =a(r-1)-5=a r^2(r-1)+4 \)
\( =a(r-1)\left(r^2-1\right)=-9 \quad \ldots \text { (ii) }\)
ii/i
\( \Rightarrow r+1=3 \)
\( \Rightarrow r=2\)
Using (i)
a(1) (1) = -3
a = -3
\(x_1=-3, x_2=-6, x_3=-12, x_4=-24 \)
\( \frac{1}{24}\left(x_1 \cdot x_2 \cdot x_3 \cdot x_4\right)=216\)
