Arithmetic Series

Pre-Calculus · Axiom Academy

One trick turns a long, tedious sum into a single multiplication. 1. A Series Is a Running Total An arithmetic sequence climbs by the same step each time — like , which adds 4 at every stage. The matching arithmetic series is just the sum of those terms. The series adds every term in the sequence Watch the brute-force way: stack the terms one at a time and let the total grow. It works, but for a hundred terms — or a thousand — it is painfully slow. Here is what Gauss noticed. Pair the first term with the last , the second with the second-to-last , and so on — folding the list in from both ends. Every pair lands on the same total. There are 50 such pairs, so the whole sum is one quick product. Watch the ends fold in and pair up: The reason every pair matches is a slick proof. Write the series forward , write it again backward , then add the two rows column by column. As one row climbs, the other falls by the same step — so every column has the identical height , a_1 + a_n . The two staircases lock into a perfect rectangle: That rectangle is n columns wide and (a_1 + a_n) tall, so it covers twice the sum: Divide by 2 — the arithmetic series formula: Often you know the common difference d rather than the last term. Substituting a_n = a_1 + (n-1)d gives the equivalent form: — average of the ends, times how many terms. — same formula, a_n written out.

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