Final answer:
The sum of the first six terms of the geometric series is calculated using the formula for the sum of a geometric series. After identifying the common ratio to be -3, the formula is applied to find the sum, which resulted in -364; however, this answer does not match the provided options.
Step-by-step explanation:
The question asks for the sum of the first six terms of a geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). To identify the common ratio for the series given (6, -18, 54, -162, ...), we divide the second term by the first term:
r = (-18) / 6 = -3.
Now, the sum Sn of the first n terms of a geometric series can be calculated using the formula
Sn = a(1 - rn) / (1 - r),
where a is the first term, r is the common ratio, and n is the number of terms. For the first six terms, we have:
a = 6, r = -3, n = 6.
Applying these values to the formula:
S6 = 6(1 - (-3)6) / (1 - (-3))
Simplifying further:
S6 = 6(1 - 729) / (1 + 3)
S6 = 6(-728) / 4
S6 = -1456 / 4
S6 = -364
However, none of the given options match this result, so it seems there may be an error either in the calculation or the options provided for the answer.