Final answer:
To place the geometric series in order from greatest to least based on their sums, calculate the sums for each series using the provided formula. The order from greatest to least is: Series A, Series B, Series D, Series C.
Step-by-step explanation:
To place the given geometric series in order from greatest to least based on their sums, we need to determine the sum of each series. The sum of a geometric series can be found using the formula:
S = (a(1 - r^n)) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Let's calculate the sums for each series:
- Series A: The first term (a) is 2 and the common ratio (r) is 2. The sum of Series A is SA = (2(1 - 2^∞)) / (1 - 2) = 2/(1-2) = 2.
- Series B: The first term (a) is 3 and the common ratio (r) is 2. The sum of Series B is SB = (3(1 - 2^∞)) / (1 - 2) = 3/(1-2) = 3.
- Series C: The first term (a) is 1 and the common ratio (r) is 5. The sum of Series C is SC = (1(1 - 5^∞)) / (1 - 5) = 1/(1-5) = 1/(-4) = -1/4.
- Series D: The first term (a) is 0.5 and the common ratio (r) is 2. The sum of Series D is SD = (0.5(1 - 2^∞)) / (1 - 2) = 0.5/(1-2) = 0.5.
Based on the sums, the order from greatest to least is:
Series A (Sum: 2)
Series B (Sum: 3)
Series D (Sum: 0.5)
Series C (Sum: -1/4).