Answer
See proof
Work Step by Step
In order to explain why $S_{n}$ of an integer arithmetic sequence is an integer we should rewrite
$S_{n}=\dfrac{a_1+a_n}{2}n$
as
$S_{n}=\dfrac{a_1+a_1+d(n-1)}{2}n$,
because $a_n=a_1+d(n-1)$.
Now
$S_n=\dfrac{2a_1+d(n-1)}{2}n=\dfrac{2a_1}{2}n+\dfrac{n(n-1)}{2}.$
$S_n=a_1n+\dfrac{n(n-1)}{2}$
The first term is obviously an integer, because $a_1$ is and integer and $n$ is always an integer. Second term is integer too, because one of the $n$ and $n-1$ must be an even number and therefore divisible by two. So $S_n$ is the sum of two integers and therefore is integer itself.
