Calculate the sum of the first six terms of a geometric series with first term 2 and common ratio 3.(1 point) Responses 186 186 242 242 364 364 728
The sum of the first six terms of a geometric series can be calculated using the formula:
\(S_n = a \left( \frac{r^n - 1}{r - 1} \right) \)
where:
\(S_n\) = sum of the first n terms of the series
\(a\) = first term
\(r\) = common ratio
\(n\) = number of terms
Given:
\(a = 2\)
\(r = 3\)
\(n = 6\)
\(S_6 = 2 \left( \frac{3^6 - 1}{3 - 1} \right) \)
\(S_6 = 2 \left( \frac{729 - 1}{2} \right) \)
\(S_6 = 2 \left( \frac{728}{2} \right) \)
\(S_6 = 2(364) \)
\(S_6 = 728 \)
Therefore, the sum of the first six terms of the geometric series is 728.