1964 AMC 12 Problem 6

For three consecutive nonzero terms, the middle term squared equals the product of its neighbors

After finding $x,$ multiply the third term by the common ratio

The geometric-mean relation gives $$(2x+2)^2=x(3x+3).$$ Under the examination’s ratio convention for a geometric progression, the consecutive ratios must be defined, so $x\ne-1.$ Dividing by $x+1$ yields $4(x+1)=3x$ and $x=-4.$ The first three terms are $-4,-6,-9,$ with ratio $\frac{3}{2}.$ The fourth is $-9(\frac{3}{2})=-\frac{27}{2}.$

Therefore, the correct answer is B.