Final answer:
The sum of the first 11 terms of the geometric sequence 2, -6, 18, -54, ... with a common ratio of -3 was incorrectly calculated as 88574 initially. The correct calculation, considering the alternating signs, should yield a sum of 88573, but this sum is not listed among the answer choices, suggesting there may be a mistake in the provided options or the computation process.
Step-by-step explanation:
To determine the sum of the first 11 terms of the geometric sequence 2, -6, 18, -54, ..., we first need to identify the common ratio (r). We can see that each term is -3 times the previous term, so the common ratio is -3. The formula for the sum of the first n terms of a geometric series is Sn = a * (1 - rn) / (1 - r), where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.
Using this formula, we calculate the sum of the first 11 terms as follows:
S11 = 2 * (1 - (-3)11) / (1 - (-3))
S11 = 2 * (1 - (-177147)) / (1 + 3)
S11 = 2 * (177148) / 4
S11 = 354296 / 4
S11 = 88574
However, we must be careful because the sequence alternates in sign and we have by mistake added the terms instead of subtracting. The correct formula considering the alternating signs is:
S11 = 2 * (1 - (-3)11) / (1 - (-3))
S11 = 2 * (1 + 177147) / 4
S11 = 2 * (177148) / 4
S11 = 88574
Because the series alternates, every pair of terms sums to -2, except the last term which stands alone. Thus, the sum is 10*(-2) + 177146 = -20 + 177146 = 177126. However, this result does not match any of the given options (a, b, c, d). Thus, there must have been a mistake in our process.
We need to re-evaluate the formula application while keeping the signs in consideration. The sum of the first 11 terms of a geometric sequence with an alternating sequence of signs is:
S11 = 2 * (-1 - (-3)11) / (1 - (-3))
S11 = 2 * (-1 + 177147) / 4
S11 = 2 * 177146 / 4
S11 = 354292 / 4
S11 = 88573
So, we conclude that there has been an error in the calculation because the correct answer should be 88573 if the given sequence and formula are applied correctly. Unfortunately, this value is not present in the provided options. Hence, we recommend the student to recheck the sequence, the formula, and the calculation to arrive at the correct answer.