Answer :
To solve the problem of finding [tex]\( S_{75} \)[/tex] for the arithmetic sequence defined by [tex]\(\{a_n\} = \{67 - 2n\}\)[/tex], follow these steps:
1. Identify the first term and the common difference:
- The sequence is given by the formula [tex]\( a_n = 67 - 2n \)[/tex].
- When [tex]\( n = 1 \)[/tex], the first term [tex]\( a_1 \)[/tex] is [tex]\( 67 - 2 \times 1 = 67 - 2 = 65 \)[/tex].
- The common difference [tex]\( d \)[/tex] can be found by considering how the sequence changes as you increase [tex]\( n \)[/tex] by 1. Since the expression is [tex]\( 67 - 2n \)[/tex], each term decreases by 2, so the common difference is [tex]\( -2 \)[/tex].
2. Calculate the 75th term ([tex]\( a_{75} \)[/tex]):
- Use the formula to find the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
- Plugging in [tex]\( n = 75 \)[/tex]:
[tex]\[
a_{75} = 65 + (75-1)(-2) = 65 + 74 \times (-2) = 65 - 148 = -83
\][/tex]
3. Calculate the sum of the first 75 terms ([tex]\( S_{75} \)[/tex]):
- Use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
[tex]\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\][/tex]
- Substituting the values:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83)) = \frac{75}{2} \times (-18) = 75 \times (-9) = -675
\][/tex]
Thus, the sum of the first 75 terms of the sequence [tex]\( S_{75} \)[/tex] is [tex]\(-675\)[/tex].
1. Identify the first term and the common difference:
- The sequence is given by the formula [tex]\( a_n = 67 - 2n \)[/tex].
- When [tex]\( n = 1 \)[/tex], the first term [tex]\( a_1 \)[/tex] is [tex]\( 67 - 2 \times 1 = 67 - 2 = 65 \)[/tex].
- The common difference [tex]\( d \)[/tex] can be found by considering how the sequence changes as you increase [tex]\( n \)[/tex] by 1. Since the expression is [tex]\( 67 - 2n \)[/tex], each term decreases by 2, so the common difference is [tex]\( -2 \)[/tex].
2. Calculate the 75th term ([tex]\( a_{75} \)[/tex]):
- Use the formula to find the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
- Plugging in [tex]\( n = 75 \)[/tex]:
[tex]\[
a_{75} = 65 + (75-1)(-2) = 65 + 74 \times (-2) = 65 - 148 = -83
\][/tex]
3. Calculate the sum of the first 75 terms ([tex]\( S_{75} \)[/tex]):
- Use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
[tex]\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\][/tex]
- Substituting the values:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83)) = \frac{75}{2} \times (-18) = 75 \times (-9) = -675
\][/tex]
Thus, the sum of the first 75 terms of the sequence [tex]\( S_{75} \)[/tex] is [tex]\(-675\)[/tex].
