Answer
Thus, for the expression $\sum\limits_{k=1}^{n}{\left( {{a}_{k}}+{{b}_{k}} \right)}=\sum\limits_{k=1}^{n}{{{a}_{k}}}+\sum\limits_{k=1}^{n}{{{b}_{k}}}$, the commutative and associative laws of addition are used.
Work Step by Step
$\sum\limits_{k=1}^{n}{\left( {{a}_{k}}+{{b}_{k}} \right)}=\sum\limits_{k=1}^{n}{{{a}_{k}}}+\sum\limits_{k=1}^{n}{{{b}_{k}}}$
$\begin{align}
& \sum\limits_{k=1}^{n}{\left( {{a}_{k}}+{{b}_{k}} \right)}=\left( {{a}_{1}}+{{b}_{1}} \right)+\left( {{a}_{1}}+{{b}_{1}} \right)+\left( {{a}_{1}}+{{b}_{1}} \right)+\ldots +\left( {{a}_{1}}+{{b}_{1}} \right) \\
& =\left( {{a}_{1}}+{{a}_{2}}+{{a}_{3}}+\ldots +{{a}_{n}} \right)+\left( {{b}_{1}}+{{b}_{2}}+{{b}_{3}}+\ldots +{{b}_{n}} \right)
\end{align}$
Using the commutative and associative laws of addition,
$\left( {{a}_{1}}+{{a}_{2}}+{{a}_{3}}+\ldots +{{a}_{n}} \right)+\left( {{b}_{1}}+{{b}_{2}}+{{b}_{3}}+\ldots +{{b}_{n}} \right)=\sum\limits_{k=1}^{n}{{{a}_{k}}}+\sum\limits_{k=1}^{n}{{{b}_{k}}}$
Obtain the left side of the expression,
$\sum\limits_{k=1}^{n}{\left( {{a}_{k}}+{{b}_{k}} \right)}=\sum\limits_{k=1}^{n}{{{a}_{k}}}+\sum\limits_{k=1}^{n}{{{b}_{k}}}$
Thus, for the expression $\sum\limits_{k=1}^{n}{\left( {{a}_{k}}+{{b}_{k}} \right)}=\sum\limits_{k=1}^{n}{{{a}_{k}}}+\sum\limits_{k=1}^{n}{{{b}_{k}}}$, the commutative and associative laws of addition are used.
