Answer
(a) geometric
(b) neither
(c) neither
(d) arithmetic
Work Step by Step
(a) Test for arithmetic sequence: $a_2-a_1=-1-1=-2$, $a_3-a_2=1-(-1)=2\ne a_2-a_1$, so it is not a arithmetic sequence.
Test for geometric sequence: $a_2/a_1=(-1)/1=-1$, $a_3/a_2=1/(-1)=-1=a_2/a_1$, $a_4/a_3=(-1)/(1)=-1=a_2/a_1$, so it is a geometric sequence and $a_5=(-1)a_4=1$
(b) Test for arithmetic sequence: $a_2-a_1=\sqrt[3] 5-\sqrt 5$, $a_3-a_2=\sqrt[6] 5-\sqrt[3] 5\ne a_2-a_1$, so it is not a arithmetic sequence.
Test for geometric sequence: $a_2/a_1=\sqrt[3] 5/\sqrt 5=(6)^{-1/6}$, $a_3/a_2=\sqrt[6] 5/\sqrt[3] 5=(5)^{-1/6}=a_2/a_1$, $a_4/a_3=1/\sqrt[6] 5=(5)^{5/6}\ne a_2/a_1$, so it is not a geometric sequence and we conclude that this sequence is neither arithmetic or geometric.
(c) Test for arithmetic sequence: $a_2-a_1=-1-2=-3$, $a_3-a_2=1/2-(-1)=3/2\ne a_2-a_1$, so it is not a arithmetic sequence. Test for geometric. we have: $a_2/a_1=(-1)/2=-1/2$, $a_3/a_2=(1/2)/(-1)=-1/2=a_2/a_1$, $a_4/a_3=2/(1/2)=4\ne a_2/a_1$, so it is not a geometric sequence, and we conclude that this sequence is neither arithmetic or geometric.
(d) Test for arithmetic sequence: $a_2-a_1=x-(x-1)=1$, $a_3-a_2=(x+1)-x=1= a_2-a_1$, $a_4-a_3=(x+2)-(x+1)=1= a_2-a_1$, so it is a arithmetic sequence, and the next term will be $a_5=a_4+1=x+3$
(There is no need to test for the geometric case, as the sequence can not be both.)
