Alright, let's solve the series step by step.
We need to find the sum of the series:
[tex]\[
\sum_{k=3}^6(-2k + 10)
\][/tex]
First, we will determine the value of each term in the series for [tex]\( k \)[/tex] from 3 to 6.
1. When [tex]\( k = 3 \)[/tex]:
[tex]\[
-2(3) + 10 = -6 + 10 = 4
\][/tex]
2. When [tex]\( k = 4 \)[/tex]:
[tex]\[
-2(4) + 10 = -8 + 10 = 2
\][/tex]
3. When [tex]\( k = 5 \)[/tex]:
[tex]\[
-2(5) + 10 = -10 + 10 = 0
\][/tex]
4. When [tex]\( k = 6 \)[/tex]:
[tex]\[
-2(6) + 10 = -12 + 10 = -2
\][/tex]
Now, we have the individual terms of the series, which are [tex]\( 4, 2, 0, \)[/tex] and [tex]\( -2 \)[/tex].
Next, we sum these terms:
[tex]\[
4 + 2 + 0 + (-2) = 4 + 2 - 2 = 4
\][/tex]
Thus, the sum of the series is:
[tex]\[
\boxed{4}
\][/tex]
