I think the answer is
$\frac{2}{3}$ or $\frac{1}{6}$
Reason
For answer 1:
Each even term is the reciprocal of the preceding odd term. If the fraction is still reducible, then the remainder is the even term.
$2/3$ is irreducible, so it is the second term. $4/3 = 1 + 1/3$. Thus, $1/3$ is the fourth term.
Similarly, $8/3 = 2 + 2/3$. So, the sixth term should be $2/3$ $$$$
For answer 2:
$A(n) = \frac{1}{2}(A(n) - 2)$ [ a simpler version posted by @Pergrine Rook. Thanks]
So we have $A(1) = 3/2 ,\ A(3) = 3/4 ,\ A(5) = 3/8,\ A(7)= 3/16\cdots $
and $A(2) =2/3 , \ A(4) =1/3, \ A(6) =1/6, \ A(8)=1/12\cdots $