A geometric sequence is generated by multiplying the first term in the sequence, [tex]a_1[/tex], several times by a fixed scalar [tex]r[/tex]. So
[tex]a_2=a_1r[/tex]
[tex]a_3=a_2r=a_1r^2[/tex]
[tex]a_4=a_3r=a_1r^3[/tex]
and so on, down to
[tex]a_n=a_{n-1}r=a_{n-2}r^2=\cdots=a_1r^{n-1}[/tex]
Using the result above, the sequences you're interested in are
31) [tex]a_n=-(-2)^{n-1}[/tex]
33) [tex]a_n=4(-4)^{n-1}[/tex]
We can rewrite these results to make them slightly cleaner.
[tex]-(-2)^{n-1}=-(-1)^{n-1}2^{n-1}=(-1)^n2^{n-1}[/tex]
[tex]4(-4)^{n-1}=4(-1)^{n-1}4^{n-1}=(-1)^{n-1}4^n[/tex]
