Find S8 for the geometric series 3 + - 6 + 12 + - 24 + ...

I guess you are asking to find the sum of the first 8 terms. If so, then:

Sum = a₁ (1-rⁿ) / (1-r), where a₁ is the 1st term, r=common ratio and n=number of terms:

the 1st term a₁ = 3

common ratio r = - 2 (since - 6/3 = - 2, and 12/-6 = - 2, etc.)

Sum = 3[ (1 - (-2) ⁸] / (1-2) = 3 (1 - 256) / (1/2)

Sum = - 1530