A question about arithmetic progressions, a.k.a.  arithmetic sequences What is the relation between the general formula of the sum of $n$ terms of an arithmetic progression, $a n^2+b n+c$, and the first term $a+b+c$ and the common difference?

If the sum of $n$ terms of an arithmetic sequence is
$$
s(n)=\color{#C00000}{an^2+bn+c}\tag{1}
$$
then $c$ is the sum of $0$ terms; that is,
$$
c=s(0)=0\tag{2}
$$
The general term of the arithmetic sequence would be
$$
\begin{align}
a(n)
&=s(n)-s(n-1)\\
&=(b-a)+(2a)n\tag{3}
\end{align}
$$
The first term of the arithmetic sequence would be
$$
a(1)=\color{#C00000}{a+b}\tag{4}
$$
Thus, the common difference of the arithmetic sequence is
$$
\begin{align}
a(n)-a(n-1)
&=((b-a)+(2a)n)-((b-a)+(2a)(n-1))\\
&=\color{#C00000}{2a}\tag{5}
\end{align}
$$
As for a relation among $(1)$, $(4)$, and $(5)$, we get
$$
\color{#C00000}{an^2+bn+c}=n(\color{#C00000}{a+b})+\frac{n^2-n}{2}\color{#C00000}{2a}\tag{6}
$$
