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Answer :
To find the sum [tex]\( S_{75} \)[/tex] of the given arithmetic sequence [tex]\(\{a_n\} = \{67 - 2n\}\)[/tex], we can follow these steps:
1. Identify the first term [tex]\( a_1 \)[/tex]:
Substitute [tex]\( n = 1 \)[/tex] into the general term formula to find the first term:
[tex]\[
a_1 = 67 - 2 \cdot 1 = 65
\][/tex]
2. Calculate the 75th term [tex]\( a_{75} \)[/tex]:
Substitute [tex]\( n = 75 \)[/tex] into the general term formula:
[tex]\[
a_{75} = 67 - 2 \cdot 75 = 67 - 150 = -83
\][/tex]
3. Use the formula for the sum of an arithmetic sequence:
The formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\][/tex]
Here, [tex]\( n = 75 \)[/tex], [tex]\( a_1 = 65 \)[/tex], and [tex]\( a_{75} = -83 \)[/tex].
4. Substitute the known values into the formula:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83))
\][/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 [tex]\( S_{75} \)[/tex] of the first 75 terms of the sequence is [tex]\(-675\)[/tex]. The correct answer is [tex]\(-675\)[/tex].
1. Identify the first term [tex]\( a_1 \)[/tex]:
Substitute [tex]\( n = 1 \)[/tex] into the general term formula to find the first term:
[tex]\[
a_1 = 67 - 2 \cdot 1 = 65
\][/tex]
2. Calculate the 75th term [tex]\( a_{75} \)[/tex]:
Substitute [tex]\( n = 75 \)[/tex] into the general term formula:
[tex]\[
a_{75} = 67 - 2 \cdot 75 = 67 - 150 = -83
\][/tex]
3. Use the formula for the sum of an arithmetic sequence:
The formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\][/tex]
Here, [tex]\( n = 75 \)[/tex], [tex]\( a_1 = 65 \)[/tex], and [tex]\( a_{75} = -83 \)[/tex].
4. Substitute the known values into the formula:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83))
\][/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 [tex]\( S_{75} \)[/tex] of the first 75 terms of the sequence is [tex]\(-675\)[/tex]. The correct answer is [tex]\(-675\)[/tex].
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