To find the sum of the series defined by
[tex]\[
\sum_{k=13}^{62}(-2k+10),
\][/tex]
we need to break down the problem step by step:
1. First, determine the first term of the sequence. The sequence follows the formula [tex]\(-2k + 10\)[/tex].
When [tex]\(k=13\)[/tex]:
[tex]\[
a = -2 \cdot 13 + 10 = -26 + 10 = -16.
\][/tex]
2. Next, determine the last term of the sequence.
When [tex]\(k=62\)[/tex]:
[tex]\[
l = -2 \cdot 62 + 10 = -124 + 10 = -114.
\][/tex]
3. Next, determine the number of terms in the sequence. The sum runs from [tex]\(k=13\)[/tex] to [tex]\(k=62\)[/tex].
The total number of terms [tex]\(n\)[/tex] is given by:
[tex]\[
n = 62 - 13 + 1 = 50.
\][/tex]
4. Finally, sum the sequence using the formula for the sum of an arithmetic series:
[tex]\[
S = \frac{n}{2} \cdot (a + l).
\][/tex]
Substituting the values we found:
[tex]\[
S = \frac{50}{2} \cdot (-16 - 114) = 25 \cdot (-130) = -3250.
\][/tex]
Thus, the sum of the series [tex]\(\sum_{k=13}^{62}(-2k+10)\)[/tex] is
[tex]\[
-3250.
\][/tex]
To summarize, the first term [tex]\(a = -16\)[/tex], the last term [tex]\(l = -114\)[/tex], the number of terms [tex]\(n = 50\)[/tex], and the sum of the series [tex]\(S = -3250\)[/tex].
