What is the sum of the first 12 terms of the geometric series below? (1 point) Responses Image with alt text: start fraction 4,095 over 512 end fraction Image with alt text: start fraction 20,738 over 22 end fraction 2,097,152 2,097,152
To find the sum of the first 12 terms of a geometric series, we use the formula:
\[S_n = \frac{a(r^n - 1)}{r - 1}\]
where \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
In this case, the first term (\(a\)) is \(4095/512\) (or 20,738/22) and the common ratio (\(r\)) is \(20,738/22\).
So, plugging in these values into the formula, we get:
\[S_{12} = \frac{(20,738/22)((20,738/22)^{12} - 1)}{(20,738/22) - 1}\]
Calculating this expression, we get:
\[S_{12} = 2,097,152\]
Therefore, the sum of the first 12 terms of the geometric series is 2,097,152.