We are given the following sequence:
[tex]a_n=\mleft\lbrace\frac{3}{4},\frac{1}{8},\frac{-1}{2},\frac{-9}{8}\mright\rbrace[/tex]
We notice that each term is determined by adding -5/8 to the previous term, like this:
[tex]\begin{gathered} \frac{3}{4}-\frac{5}{8}=\frac{1}{8} \\ \\ \frac{1}{8}-\frac{5}{8}=-\frac{1}{2} \\ \\ -\frac{1}{2}-\frac{5}{8}=-\frac{9}{8} \end{gathered}[/tex]
Therefore, the sequence is an arithmetic sequence with a common difference of -5/8.
Arithmetic sequences are convergent only when the common diference is zero, therfore, the sequence is divergent.
