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.

Question

(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 \(\boxed{-675}\).