Answer:
B
Step-by-step explanation:
We are given that:
[tex]8x-3, \, 4x+1, \text{ and } 2x-2[/tex]
Are consecutive terms of an arithmetic sequence.
And we want to determine the common difference d.
Recall that for an arithmetic sequence, each subsequent term is d more than the previous term.
In other words, the second term is one d more than the first term. So:
[tex]4x+1=(8x-3)+d[/tex]
And the third term is two d more than the first term. So:
[tex]2x-2=(8x-3)+2d[/tex]
We can isolate the d in the first equation:
[tex]-4x+4=d[/tex]
As well as the second:
[tex]-6x+1=2d\Rightarrow \displaystyle -3x+\frac{1}{2}=d[/tex]
Then by substitution:
[tex]\displaystyle -4x+4=-3x+\frac{1}{2}[/tex]
Solve for x:
[tex]\displaystyle -x=\frac{-7}{2}\Rightarrow x=\frac{7}{2}[/tex]
The isolated first equation tells us that:
[tex]d=-4x+4[/tex]
Therefore:
[tex]\begin{aligned} \displaystyle d&=-4\left(\frac{7}{2}\right)+4\\&=-2(7)+4\\&=-14+4\\&=-10 \end{aligned}[/tex]
Our final answer is B.
