In an arithmetic sequence whose 4th term is 14 and whose 11th term is 70, the sum of the first 12 terms of the sequence is 15.
Explanation
To find the sum of the first 12 terms of the arithmetic sequence, we need to use the formula for the sum of the first \( n \) terms of an arithmetic sequence:
\[
S_n = \frac{n}{2} \times (2a + (n - 1) \times d)
\]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( S_n \) is the sum of the first \( n \) terms.
From the previous calculations, we found:
- The first term \( a = -10 \),
- The common difference \( d = 8 \).
Given \( n = 12 \), we need to find \( S_{12} \).
Step-by-Step Calculation:
1. Substitute the values into the sum formula:
\[
S_{12} = \frac{12}{2} \times (2a + (12 - 1) \times d)
\]
\[
S_{12} = 6 \times (2 \times (-10) + 11 \times 8)
\]
2. Calculate inside the parentheses:
\[
2 \times (-10) = -20
\]
\[
11 \times 8 = 88
\]
\[
2a + (n - 1) \times d = -20 + 88 = 68
\]
3. Calculate the sum:
\[
S_{12} = 6 \times 68 = 408
\]
The sum of the first 12 terms of the sequence is 408.
The correct answer is D. 408.