To find the sum of the arithmetic sequence given by [tex]\(\sum_{n=0}^{28} (7 - 5n)\)[/tex], we'll follow these steps:
1. Identify the first term ([tex]\(a_0\)[/tex]) and the common difference ([tex]\(d\)[/tex]).
The general form of the sequence is [tex]\(a_n = 7 - 5n\)[/tex].
- The first term is when [tex]\(n = 0\)[/tex]:
[tex]\[
a_0 = 7 - 5 \cdot 0 = 7
\][/tex]
- The common difference ([tex]\(d\)[/tex]) is the difference between any two consecutive terms:
[tex]\[
d = a_{n+1} - a_n = (7 - 5(n+1)) - (7 - 5n) = -5
\][/tex]
2. Determine the nth term of the sequence, particularly the last term ([tex]\(a_{28}\)[/tex]).
Using the formula for the nth term of the sequence [tex]\(a_n = a_0 + n \cdot d\)[/tex]:
[tex]\[
a_{28} = 7 + 28 \cdot (-5) = 7 - 140 = -133
\][/tex]
3. Find the sum of the arithmetic sequence from [tex]\(n = 0\)[/tex] to [tex]\(n = 28\)[/tex].
The formula for the sum of the first [tex]\(n + 1\)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{(n+1)}{2} \cdot (a_0 + a_n)
\][/tex]
Substituting the values we have:
[tex]\[
S_{28} = \frac{28 + 1}{2} \cdot (7 + (-133))
\][/tex]
4. Calculate the sum.
Simplifying inside the parentheses first:
[tex]\[
7 + (-133) = 7 - 133 = -126
\][/tex]
Now, perform the calculation:
[tex]\[
S_{28} = \frac{29}{2} \cdot (-126)
\][/tex]
Multiplying the values:
[tex]\[
S_{28} = \frac{29 \cdot (-126)}{2} = \frac{-3654}{2} = -1827
\][/tex]
Therefore, the sum of the arithmetic sequence [tex]\(\sum_{n=0}^{28}(7 - 5n)\)[/tex] is [tex]\(-1827.0\)[/tex].
