Final answer:
To evaluate the geometric series 54 + 18 + 6 + 2 + ... for 100 terms, we use the formula Sn = a1 * (1 - rn) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms. After calculation, the sum S100 ≈ 54 * (3/2) = 81, which does not match the given options.
Step-by-step explanation:
To evaluate the geometric series for the given number of terms, we can use the formula for the sum of a finite geometric series:
Sn = a1 * (1 - rn) / (1 - r), where
- a1 is the first term of the series,
- r is the common ratio between the terms,
- n is the number of terms.
Given the series 54 + 18 + 6 + 2 + ..., we can see:
- a1 = 54,
- The common ratio r = 18/54 = 1/3,
- n = 100 terms.
Plugging these values into the formula:
S100 = 54 * (1 - (1/3)100) / (1 - 1/3) = 54 * (1 - (1/3)100) / (2/3)
This simplifies to:
S100 = 54 * (3/2) * (1 - (1/3)100)
Since (1/3)100 is an exceedingly small number, it approaches 0 when considering the first 100 terms of the series. Therefore, the sum of the 100 terms approximately equals:
S100 ≈ 54 * (3/2) = 81.
However, this does not match with any of the provided options, so it seems there might be an error in the options.