Question

Find S12
for the arithmetic sequence 7,5,3,1,−1,−3,….

S12=

Answer 1

To find \( S_{12} \), the sum of the first 12 terms of the arithmetic sequence \( 7, 5, 3, 1, -1, -3, \ldots \), we first need to identify the first term and the common difference.

1. **Identify the first term and common difference:**
- The first term \( a = 7 \).
- The common difference \( d = 5 - 7 = -2 \).

2. **Use the formula for the sum of the first \( n \) terms of an arithmetic sequence:**
\[
S_n = \frac{n}{2} \times (2a + (n - 1)d)
\]

3. **Substituting the values to find \( S_{12} \):**
- Here, \( n = 12 \), \( a = 7 \), and \( d = -2 \).
\[
S_{12} = \frac{12}{2} \times (2 \times 7 + (12 - 1)(-2))
\]

4. **Calculate the components:**
- Calculate \( 2a \):
\[
2 \times 7 = 14
\]
- Calculate \( (n - 1)d \):
\[
(12 - 1)(-2) = 11 \times (-2) = -22
\]

5. **Combine these values:**
\[
2a + (n - 1)d = 14 - 22 = -8
\]

6. **Substitute back to find \( S_{12} \):**
\[
S_{12} = \frac{12}{2} \times (-8) = 6 \times (-8) = -48
\]

Thus, the sum of the first 12 terms \( S_{12} \) is:
\[
\boxed{-48}
\]