Gauss's Method

Algebra 2 · Axiom Academy

The pairing trick a ten-year-old used to add 1 to 100 in seconds — and how it becomes the formula for any arithmetic series. Instead of grinding left to right, Gauss folded the list in half. He paired the first number with the last, the second with the second-to-last, and so on: 1 with 100, 2 with 99, 3 with 98. Every pair adds up to the same total, 101. The tedious sum Gauss was handed Fold the ends together — every pair makes 101 2. Why the Pairs Are Always Equal To see that this never fails, write the sum S forwards, then write it again backwards underneath. Now add the two rows column by column. Each column pairs a term with its mirror. Because the list climbs in equal steps, whatever the top term gains the bottom term loses — so every column adds to the very same value , the first term plus the last. There are n columns and each equals the same total, so the two rows together — that is, twice the sum — come to: 3. The Arithmetic-Series Formula We found that twice the sum equals n times the first-plus-last. Halving both sides gives the formula for the sum of any arithmetic series: A picture makes it undeniable. Twice the sum is a rectangle n wide and first-plus-last tall. The series itself is a staircase that fills exactly half of that rectangle — the other half is the same staircase turned upside down. For the classic run 1, 2, 3, …, n , the first term is 1 and the last is n , so first-plus-last is n + 1 and the formula collapses to the famous triangular-number result:

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