Answer :
To find [tex]\( S_{75} \)[/tex], the sum of the first 75 terms of the arithmetic sequence defined by [tex]\( a_n = 67 - 2n \)[/tex], let's follow these steps:
1. Identify the first term and the 75th term:
- The first term [tex]\( a_1 \)[/tex] of the sequence is when [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 67 - 2 \times 1 = 65 \][/tex]
- The 75th term [tex]\( a_{75} \)[/tex] is when [tex]\( n = 75 \)[/tex]:
[tex]\[ a_{75} = 67 - 2 \times 75 = -83 \][/tex]
2. Use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
The formula for the sum [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = \frac{n}{2} \times (a_1 + a_n) \][/tex]
For this sequence, [tex]\( n = 75 \)[/tex], [tex]\( a_1 = 65 \)[/tex], and [tex]\( a_{75} = -83 \)[/tex]. Plug these values into the formula:
[tex]\[ S_{75} = \frac{75}{2} \times (65 + (-83)) \][/tex]
3. Calculate [tex]\( S_{75} \)[/tex]:
[tex]\[ S_{75} = \frac{75}{2} \times (65 - 83) \][/tex]
[tex]\[ S_{75} = \frac{75}{2} \times (-18) \][/tex]
[tex]\[ S_{75} = 75 \times (-9) \][/tex]
[tex]\[ S_{75} = -675 \][/tex]
Therefore, the sum of the first 75 terms of the sequence is [tex]\( -675 \)[/tex].
1. Identify the first term and the 75th term:
- The first term [tex]\( a_1 \)[/tex] of the sequence is when [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 67 - 2 \times 1 = 65 \][/tex]
- The 75th term [tex]\( a_{75} \)[/tex] is when [tex]\( n = 75 \)[/tex]:
[tex]\[ a_{75} = 67 - 2 \times 75 = -83 \][/tex]
2. Use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
The formula for the sum [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = \frac{n}{2} \times (a_1 + a_n) \][/tex]
For this sequence, [tex]\( n = 75 \)[/tex], [tex]\( a_1 = 65 \)[/tex], and [tex]\( a_{75} = -83 \)[/tex]. Plug these values into the formula:
[tex]\[ S_{75} = \frac{75}{2} \times (65 + (-83)) \][/tex]
3. Calculate [tex]\( S_{75} \)[/tex]:
[tex]\[ S_{75} = \frac{75}{2} \times (65 - 83) \][/tex]
[tex]\[ S_{75} = \frac{75}{2} \times (-18) \][/tex]
[tex]\[ S_{75} = 75 \times (-9) \][/tex]
[tex]\[ S_{75} = -675 \][/tex]
Therefore, the sum of the first 75 terms of the sequence is [tex]\( -675 \)[/tex].
