A geometric series a_n starts from a certain number a_1=x and computes the next term a_{n+1} by multiplying the previous one by a fixed ratio r. So, you have
a_1=x
a_2=a_1*r=x*r
a_3 = a_2*r = (x*r)*r=x*r^2
a_4 = a_3*r = (x*r^2)*r=x*r^3
So, it should be clear that the general rule is
a_n = x*r^{n-1}
So, to find the generi expression, we need both x and r. Note that, since we have the first term in each series, we immediately know x: you just need to look at the first term, since a_1=x.
So, if we call the first series a_n and the second b_n, we have a_1=4 and b_1=16.
To compute r, we can divide two consecutive terms, since you have
a_{n+1}/a_n = (x*r^n)/(x*r^{n-1})=r
So, if we call r_a the ratio of the first series and r_b the ratio of the second series, and we use the first two terms to compute it, we have
r_a = a_2/a_1=-4/4=-1,\qquad r_b=b_2/b_1=8/16=1/2
So, we have the general rules:
a_n = 4*(-1)^{n-1},\qquad b_n = 16*(1/2)^{n-1} = 2^4*2^{-n+1} = 2^{5-n}
To compute the eight element, simply plug n=8:
a_8 = 4*(-1)^7 = 4*(-1) = -4
b_8 = 2^{5-8} = 2^{-3} = 1/8
