Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 845: 84

$x=2$

A sequence is geometric if there exists a common ratio $r$ among consecutive terms such as: $r=\dfrac{a_n}{a_{n−1}}$ In order for a sequence to be geometric, the quotient of all consecutive terms must remain constant. So, we have: $\dfrac{a_3}{a_2}=\dfrac{a_2}{a_1}$. Substituting the given values of the first three terms of the sequence gives: $$\dfrac{x+2}{x}=\dfrac{x}{x-1}$$ Next, we will do cross-multiplication, then solve for $x$ to obtain: \begin{align*}(x+2)(x-1)&=(x)(x)\\ x(x-1)+2(x-1)&=x^2\\ x^2-x+2x-2&=x^2\\ x^2+x-2&=x^2\\ x^2+x-2-x^2&=x^2-x^2\\ x-2&=0\\ x&=2\end{align*}