Answer :
To calculate [tex]\( S_{75} \)[/tex], the sum of the first 75 terms of the arithmetic sequence defined by [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 formula for the sequence:
[tex]\[
a_1 = 67 - 2 \times 1 = 65
\][/tex]
2. Find the common difference ([tex]\(d\)[/tex]):
The common difference is the change in the sequence as [tex]\( n \)[/tex] increases by 1. From the sequence formula [tex]\( a_n = 67 - 2n \)[/tex], the difference between consecutive terms is:
[tex]\[
d = -2
\][/tex]
3. Find the 75th term ([tex]\(a_{75}\)[/tex]):
Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Plug in the values:
[tex]\[
a_{75} = 65 + (75 - 1) \cdot (-2) = 65 - 148 = -83
\][/tex]
4. 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]
Substitute the values we've found:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83)) = \frac{75}{2} \times (-18)
\][/tex]
[tex]\[
S_{75} = 75 \times (-9) = -675
\][/tex]
Therefore, the sum [tex]\( S_{75} \)[/tex] is [tex]\(-675\)[/tex].
1. Identify the first term ([tex]\(a_1\)[/tex]):
Substitute [tex]\( n = 1 \)[/tex] into the formula for the sequence:
[tex]\[
a_1 = 67 - 2 \times 1 = 65
\][/tex]
2. Find the common difference ([tex]\(d\)[/tex]):
The common difference is the change in the sequence as [tex]\( n \)[/tex] increases by 1. From the sequence formula [tex]\( a_n = 67 - 2n \)[/tex], the difference between consecutive terms is:
[tex]\[
d = -2
\][/tex]
3. Find the 75th term ([tex]\(a_{75}\)[/tex]):
Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Plug in the values:
[tex]\[
a_{75} = 65 + (75 - 1) \cdot (-2) = 65 - 148 = -83
\][/tex]
4. 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]
Substitute the values we've found:
[tex]\[
S_{75} = \frac{75}{2} \times (65 + (-83)) = \frac{75}{2} \times (-18)
\][/tex]
[tex]\[
S_{75} = 75 \times (-9) = -675
\][/tex]
Therefore, the sum [tex]\( S_{75} \)[/tex] is [tex]\(-675\)[/tex].
