$($Optional$.$ Bonus $+2$ points if you answer it correctly$)$ Much like a geometric series, an arithmetic series is a series ${\displaystyle \underset{k=1}{\overset{}{}}}{a}_k$ in which each term is computed from the previous one by adding $($or subtracting$)$ a constant $r.$ More precisely, ${\displaystyle \underset{k=1}{\overset{}{}}}{a}_k$ is an arithmetic series if there's a constant $r$ such that $a_{k+1}=a_k+r$ for all $k($or $r=a_{k+1}-a_k).$

$($a$)$ Write out the first four terms of the series $($that is$,a_1,a_2,a_3,$ and $a_4)$ in terms of $a_1,$ and $r.$

$($b$)$ Write out the $n$th partial sum $S_n={\displaystyle \underset{k=1}{\overset{n}{}}}{a}_k$ in cdots notation in terms of $a_1,n$ and $r$ only.

$($c$)$ Knowing that ${\displaystyle \underset{k=1}{\overset{n}{}}}k=1+2+$cdots$+n=\frac{n(n+1)}{2},$ find a closed formula for the $n$th partial sum $S_n={\displaystyle \underset{k=1}{\overset{n}{}}}{a}_k.$