$\displaystyle \frac{e}{2},0,-\frac{e}{2},-\frac{3e}{2}$

$\displaystyle a_2-a_1=-\frac{e}{2}$

$\displaystyle a_3-a_2=-\frac{e}{2}$

$\displaystyle a_4-a_3=-\frac{3e}{2}-(-\frac{e}{2})=-\frac{2e}{2}=-e$

There seems to be problem with the common difference here. The sequence comes from the following study guide, example 4:

College Algebra Tutorial on Arithmetic Sequences and Series

It doesn't seem like this is an arithmetic sequence to me.