To find the sum of the first 12 terms ([tex]\( S_{12} \)[/tex]) of the given arithmetic series [tex]\(-5, -14, -23, -32, \ldots\)[/tex], we need to use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic series:
[tex]\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \][/tex]
Where:
- [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms.
- [tex]\( n \)[/tex] is the number of terms.
- [tex]\( a \)[/tex] is the first term of the series.
- [tex]\( d \)[/tex] is the common difference between consecutive terms.
Let’s proceed step-by-step.
1. Identify the first term ([tex]\( a \)[/tex]):
The first term of the series is [tex]\( a = -5 \)[/tex].
2. Find the common difference ([tex]\( d \)[/tex]):
To find the common difference, subtract the first term from the second term:
[tex]\[ d = -14 - (-5) = -14 + 5 = -9 \][/tex]
3. Determine the number of terms ([tex]\( n \)[/tex]) to be summed:
We need to find the sum of the first 12 terms, so [tex]\( n = 12 \)[/tex].
4. Substitute [tex]\( a \)[/tex], [tex]\( d \)[/tex], and [tex]\( n \)[/tex] into the sum formula:
[tex]\[ S_{12} = \frac{12}{2} \left( 2(-5) + (12-1)(-9) \right) \][/tex]
5. Simplify inside the parentheses first:
[tex]\[ S_{12} = 6 \left( -10 + 11(-9) \right) \][/tex]
6. Calculate the expression inside the parentheses:
[tex]\[ -10 + 11(-9) = -10 - 99 = -109 \][/tex]
7. Now multiply by 6 to get the final sum:
[tex]\[ S_{12} = 6 \times (-109) = -654 \][/tex]
The sum of the first 12 terms of the series is [tex]\( S_{12} = -654 \)[/tex], which matches one of the given choices.
So, the correct answer is:
[tex]\[ \boxed{-654} \][/tex]
