To find the sum of the series
[tex]\[ \sum_{k=3}^6 (-2k + 5), \][/tex]
we need to evaluate the expression [tex]\(-2k + 5\)[/tex] for each integer value of [tex]\(k\)[/tex] from 3 to 6 inclusive, and then sum these values.
1. When [tex]\( k = 3 \)[/tex]:
[tex]\[ -2(3) + 5 = -6 + 5 = -1 \][/tex]
2. When [tex]\( k = 4 \)[/tex]:
[tex]\[ -2(4) + 5 = -8 + 5 = -3 \][/tex]
3. When [tex]\( k = 5 \)[/tex]:
[tex]\[ -2(5) + 5 = -10 + 5 = -5 \][/tex]
4. When [tex]\( k = 6 \)[/tex]:
[tex]\[ -2(6) + 5 = -12 + 5 = -7 \][/tex]
Now, we sum these four results:
[tex]\[ -1 + (-3) + (-5) + (-7) \][/tex]
[tex]\[ = -1 - 3 - 5 - 7 \][/tex]
[tex]\[ = -16 \][/tex]
Therefore, the sum of the series is:
[tex]\[
\boxed{-16}
\][/tex]
