Arithmetic Series Sum

Write the series out, then write it out again underneath in reverse order, and add the two lines column by column. Every column comes to the same thing — the first term plus the last — because whatever one line gains going right, the other loses. There are $n$ columns and you have counted the series twice, so $2S = n(a_1 + a_n)$, which is $S_n = \tfrac{n}{2}(a_1 + a_n)$. Unpack the last term as $a_1 + (n-1)d$ and you have the form on this page. The trick is credited to Gauss at about age nine, told to add the whole numbers up to 100 and answering 5050 almost immediately, and whether or not the story is true it is the right way to see the formula.

Read the result plainly and it says: the sum is the number of terms times the average term, and for an arithmetic sequence the average is simply the midpoint between the first and the last. No other kind of sequence lets you get away with that.

Stacking problems fall out directly. A pile of pipe with 15 on the bottom row and one fewer per row for 10 rows holds $\tfrac{10}{2}(2 \times 15 + 9 \times (-1)) = 105$ lengths. An amphitheatre with 20 seats in the front row and three more in each row behind it seats $\tfrac{12}{2}(40 + 11 \times 3) = 438$ people over twelve rows. The solver also runs backwards, recovering the first term or the step from a known total.

The mistake that dominates this topic is the $(n-1)$. Twelve rows have only eleven gaps between them, so the twelfth row is eleven steps up from the first, not twelve. It is the same off-by-one that makes a 100 m fence with posts every 10 m need eleven posts, and it survives into professional work because the wrong answer is close enough to look right. A related version: summing from the 5th term to the 20th covers 16 terms, not 15, because both ends count. Two smaller traps. The step $d$ is signed — the pipe stack above needs $d = -1$, and entering 1 describes a pile that widens as it rises. And $n$ has to be a whole positive count of terms; a fractional answer when solving backwards means no such series exists. Finally, notice the family resemblance to the trapezoid: average of the two ends times the extent. An arithmetic series is a trapezoid counted in discrete steps, which is a fair way to remember either formula from the other.