Question

Calculate upper S subscript 75 for the arithmetic sequence defined by left-brace a subscript n baseline right-brace equals left-brace 67 minus 2 n right-brace.
(1 point)
Responses

83
83

negative 83
Image with alt text: negative 83

negative 675
Image with alt text: negative 675

negative 1,350
Image with alt text: negative 1,350

Answer 1

To find the sum \( S_{75} \) of the first 75 terms of the arithmetic sequence defined by \( a_n = 67 - 2n \), we can follow these steps:

1. **Identify the first term \( a_1 \) and the 75th term \( a_{75} \):**
- The first term \( a_1 \):
\[
a_1 = 67 - 2(1) = 67 - 2 = 65
\]
- The 75th term \( a_{75} \):
\[
a_{75} = 67 - 2(75) = 67 - 150 = -83
\]

2. **Determine the number of terms \( n \):**
- Here, \( n = 75 \).

3. **Calculate the sum of the first \( n \) terms \( S_n \) using the formula for the sum of an arithmetic series:**
- The formula is:
\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]
- Plugging in the values we have:
\[
S_{75} = \frac{75}{2} \times (65 + (-83))
\]
\[
S_{75} = \frac{75}{2} \times (65 - 83)
\]
\[
S_{75} = \frac{75}{2} \times (-18)
\]
\[
S_{75} = \frac{75 \times (-18)}{2} = \frac{-1350}{2} = -675
\]

Thus, the value of \( S_{75} \) is \(\boxed{-675}\).