Rows in Pascal's Triangle containing Numbers in Arithmetic Progression

From ProofWiki
Jump to navigation Jump to search

Theorem

There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic progression.


Proof


Examples

$7, 21, 35$

The integers:

$7, 21, 35$

are in arithmetic progression and appear in row $7$ of Pascal's triangle.


$1001, 2002, 3003$

The integers:

$1001, 2002, 3003$

are in arithmetic progression and appear in row $14$ of Pascal's triangle.


$490 \,314, 817 \, 190, 1 \, 144 \, 066$

The integers:

$490 \, 314, 817 \, 190, 1 \, 144 \, 066$

are in arithmetic progression and appear in row $23$ of Pascal's triangle.


Also see


Historical Note

This result, along with Rows in Pascal's Triangle containing Numbers in Geometric Progression and Rows in Pascal's Triangle containing Numbers in Harmonic Progression, was apparently published by Theodore Samuel Motzkin in Volume $12$ of Scripta Mathematica, but details have not been established.


Sources