Answer:
To find the sum of the first 14 terms of the series, we can observe that each term is part of an arithmetic sequence with a common difference of -8.
The general form of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.
In this series:
- \(a_1 = 12\) (the first term)
- \(d = -8\) (the common difference)
The sum of the first \(n\) terms of an arithmetic series is given by the formula \(S_n = \frac{n}{2}(a_1 + a_n)\).
Let's calculate it for \(n = 14\):
\[ S_{14} = \frac{14}{2}(12 + a_{14}) \]
\[ a_{14} = a_1 + (14-1)d \]
\[ a_{14} = 12 + (13)(-8) = -92 \]
Now substitute this into the sum formula:
\[ S_{14} = \frac{14}{2}(12 + (-92)) \]
\[ S_{14} = \frac{14}{2}(-80) = -560 \]
So, the correct answer is:
C. -560