Final answer:
The sum of the first 11 terms of the geometric sequence 2, -6, 18, -54, ... is -171590. This is calculated using the formula for the sum of a geometric series, with the common ratio of -3 and 11 terms.
Step-by-step explanation:
To determine the sum of the first 11 terms of the geometric sequence 2, –6, 18, –54, ..., we need to use the formula for the sum 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.
The common ratio (r) in this sequence can be found by dividing the second term by the first term: r = –6 / 2 = –3. Now we can apply the sum formula for a geometric series, which is Sn = a(1 - rn) / (1 - r), where Sn is the sum of the first n terms, a is the first term in the series, and n is the number of terms to sum.
For the first 11 terms, the formula becomes S11 = 2(1 - (–3)11) / (1 - (–3)). Calculating the powers and dividing, we find that S11 = 2(1 - (–171591)) / (1 + 3) = 2(–171590) / 4 = –171590. Therefore, the sum of the first 11 terms of the given geometric sequence is –171590.