Question

Find x so that x + 3, 2x + 1, and 5x + 2 are consecutive terms of an arithmetic sequence.

Short answer

The value of x is$-\frac{3}{2}$

Step-by-step solution

Given Information 

x + 3, 2x + 1, and 5x + 2 are consecutive terms of an arithmetic sequence

common difference 

$$\begin{gathered} a_1=x+3 \\ {a}_2=2x+1 \\ {a}_3=5x+2 \\ {a}_2-a_1=2x+1-(x+3)=x-2 \\ {a}_3-a_2=5x+2-(2x+1)=3x+1 \end{gathered}$$

 common difference is a constant 

common difference is constant in arithmetic sequence

$$\begin{gathered} a_2-a_1=a_3-a_2 \\ x-2=3x+1 \\ x-3x-2=1 \\ -2x-2=1 \\ -2x=1+2 \\ -2x=3 \\ x=-\frac{3}{2} \\ thevalueofx=-\frac{3}{2} \end{gathered}$$