Finite differences of arithmetic partial sums ($S_{n+1}-S_n=a_{n+1},{\Delta }^2{S}_n=d$)

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 givesThus 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 $S_n=n{a}_1$.