Answer
True.
Work Step by Step
$\displaystyle \sum_{i=1}^{n}(a_{i}+b_{i})=$
$=(a_{1}+b_{1})+(a_{2}+b_{2})+......+(a_{n}+b_{n})$
... addition is associative and commutative
... (we can remove the parentheses and group terms in any order)
$=(a_{1}+a_{2}+...+a_{n})+(b_{1}+b_{2}+...+b_{n})$
... both parentheses can be written in summation notation
$=(\displaystyle \sum_{i=1}^{n}a_{i})+(\sum_{i=1}^{n}b_{i})$
