What is the sum of the geometric series Capital sigma, subscript n equals 1, superscript 10, baseline 6 times 2 superscript n.?
First, let's express the geometric series in general form:
\[a_r = 6 \cdot 2^r\]
where \(r\) is the term number starting from 1.
Now we can find the sum of the first 10 terms of the series using the formula for the sum of a geometric series:
\[S_n = a \frac{r^n - 1} {r - 1}\]
where:
- \(S_n\) is the sum of the first \(n\) terms
- \(a\) is the first term of the series
- \(r\) is the common ratio
- \(n\) is the number of terms
In this case, the first term \(a = a_1 = 6 \cdot 2^1 = 12\) and the common ratio \(r = 2\).
Therefore, the sum of the first 10 terms of the series is:
\[S_{10} = 12 \frac{2^{10} - 1} {2 - 1} = 12 \cdot \frac{1023} {1} = 12216\]
So, the sum of the geometric series \( \sum_{n=1}^{10} 6 \cdot 2^n \) is 12216.