Final answer:
The sum of the first 12 terms of the given geometric series, calculated using the formula for the sum of a geometric series, results in -20132658, which is not listed in the provided answer choices. A possible error in the question or answer choices was identified.
Step-by-step explanation:
The question asks for the sum of the first 12 terms (S_12) of the geometric series 3 + (-12) + 48 + (-192) + ... Since it's a geometric series, each term is multiplied by a common ratio (r) to get the next term. In this case, the common ratio is -4: (3 * -4 = -12), (-12 * -4 = 48), and so on.
To find the sum of a geometric series, we use the formula S_n = a(1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a is the first term of the series, and n is the number of terms. Here, a = 3, r = -4, and n = 12. We substitute these into the formula:
S_12 = 3(1 - (-4)^12) / (1 - (-4))
Calculating the value, we get S_12 = 3(1 - 16777216) / 5 = 3(-16777215) / 5 = -100663290 / 5 = -20132658, which is not an option provided in the question. Hence, there might be an error in the question or the answer choices. However, the closest answer choice, assuming a sign error, would be (b) -10066329.