Proof without Words: 1+2+...+(n-1)=n Choose 2
Ok, just in case, here are some words: The number of subsets of size two taken from a set of size is denoted by , read as choose 2. This Demonstration shows that for each such subset (shown as two blue disks in the bottom row) there is a unique orange disk in the triangle above. Since there are disks in this triangle (one summand for each row), this sum must be equal to . Click any disk above the bottom row to explore this association.
Inspired by R. B. Nelson, "Visual Gems of Number Theory," Math Horizons, February, 2008.