Sum of n Terms of an AP

The sum of the first $n$ terms of an AP, $S_n$, has a direct closed-form formula — useful whenever a running total is needed (total savings over several months, total output over several years).

Where it comes from (the pairing idea): write the sum forwards and backwards and add — the first and last term give $a + l$, the second and second-last also give $a + l$, and so on. There are $n$ such equal pairs across the two copies, so $2S_n = n(a+l)$, giving $S_n = \tfrac{n}{2}(a+l)$. (This is the reasoning famously attributed to the young Gauss.)

Example: the sum of the first $20$ terms of $4, 7, 10, \ldots$ ($a = 4,\ d = 3$) is

Checked by the other form: the $20$th term is $l = 4 + 19(3) = 61$, so $S_{20} = \tfrac{20}{2}(4 + 61) = 10(65) = 650$ — matches.