Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.2 Arithmetic Sequences - 11.2 Assess Your Understanding - Page 835: 55

$x=\dfrac{-3}{2}$

In order to make the terms $x+3$, $2x+1$ and $5x+2$ consecutive terms in an arithmetic sequence, we must have a common difference between the terms. For this, we have: $(2x+1)-(x+3) =(5x+2) -(2x+1)$ This yields: $x-2=3x+1$ or, $2x=-3$ or, $x=\dfrac{-3}{2}$