Answer
See explanation.
Work Step by Step
Rewrite sum
\[
\sum_{k=1}^{n}\left(-b_{k}+a_{k}\right)=\sum_{k=1}^{n}\left(a_{k}+\left(-b_{k}\right)\right)
\]
Using theorem 5.4.1
$\sum_{k=1}^{n}\left(-b_{k}+a_{k}\right)=\sum_{k=1}^{n} a_{k}+\sum_{k=1}^{n}\left(-b_{k}\right)$
Using theorem 5.4.1 (a) with $c=-1$
(b)
$\sum_{k=1}^{n}\left(-b_{k}+a_{k}\right)=-\sum_{k=1}^{n} b_{k}+\sum_{k=1}^{n} a_{k}$
