Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 825: 83

$x=-4$

In order for a sequence to be geometric, the quotient of all consecutive terms must be constant. Hence here: $\frac{a_3}{a_2}=\frac{a_2}{a_1}\\\frac{x+3}{x+2}=\frac{x+2}{x}\\x(x+3)=(x+2)(x+2)\\x^2+3x=x^2+4x+4\\3x=4x+4\\-x=4\\x=-4$