Arithmetic Series

Algebra 1 · Axiom Academy

Add a long run of evenly spaced numbers in one stroke — the pairing trick a young Gauss used to find in seconds. Suppose we want the sum of the first ten positive integers. We could grind it out one step at a time — but watch how the running total climbs, and ask whether there's a shortcut hiding in the shape. Here is Gauss's idea. Pair the first term with the last , the second with the second-to-last , and keep folding inward. Every pair lands on the same total. Ten terms fold into five pairs, and each pair sums to 11 . So the whole sum is just . 3. Write It Forwards and Backwards Why does the pairing always balance? Write the sum once left-to-right, then again right-to-left directly beneath it. Now add the two rows column by column . Each column holds a first-row term and the term it's paired with — so every column adds to the same a_1+a_n . With n columns, the two rows together make n(a_1+a_n) . But that double-counts the sum, so we halve it. 4. Gauss and the Numbers 1 to 100 This is the legend itself. A young Gauss, told to add every number from 1 to 100 , folded the list in half: 1 with 100 , 2 with 99 , and so on — fifty pairs, each summing to 101 . Here n=100 , a_1=1 , and a_n=100 . The formula does the same fold in symbols: Nothing here was special to 1 through 10 . Take any arithmetic sequence — first term a , common difference d — and the pairing balances for the same reason. Each term is d larger than the one before it.

This is the written version of the interactive lesson above. See the full Algebra 1 course.