= Finite differences of arithmetic partial sums
{title2=$S_{n+1}-S_n=a_{n+1},\quad\Delta^2S_n=d$}
Adding one <sequence term> changes a <partial sum> by exactly that entry, so $S_{n+1}-S_n=a_{n+1}$. Taking another <finite difference> gives
$$
S_{n+2}-2S_{n+1}+S_n=a_{n+2}-a_{n+1}=d.
$$
Thus arithmetic <sequence terms> produce <partial sums> with constant second <finite differences>. Conversely, if these second <finite differences> are a fixed $d$ and $a_{n+1}=S_{n+1}-S_n$, then those entries form an <arithmetic progression>. For nonzero $d$, the displayed sum formula is a <quadratic function>; for $d=0$, it reduces to the linear expression $S_n=na_1$.
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