Legend says young Gauss summed 1 to 100 in seconds: pair the first with the last (101), the second with the second-last (101 again), and so on — fifty identical pairs, $50 \times 101 = 5050$. That pairing IS the formula: $S_n = \tfrac{n}{2}\left(2a + (n - 1)\,d\right)$ — read aloud: S-n equals n over two, times two-a plus n-minus-one d. The $\tfrac{n}{2}$ counts the pairs; the bracket is first term plus last term.

Where you will meet it: theatre rows widening toward the back, pipes stacked in narrowing rows, posts, seats, seconds. The resident traps are a matched pair: multiplying by $n$ as if every row matched the first (or the last), and dropping the $\tfrac{n}{2}$'s division — Gauss counts every pair once, never twice.