Proof without Words: 1+2+...+(n-1)=n Choose 2
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.
n
n |
2 |
n
1+2+⋯+(n-1)
n |
2 |
Details
Details
Inspired by R. B. Nelson, "Visual Gems of Number Theory," Math Horizons, February, 2008.
External Links
External Links
Permanent Citation
Permanent Citation
Bruce Torrence
"Proof without Words: 1+2+...+(n-1)=n Choose 2"
http://demonstrations.wolfram.com/ProofWithoutWords12N1NChoose2/
Wolfram Demonstrations Project
Published: March 7, 2011