Pascal's Triangle Properties

Combinatorics · Axiom Academy

LESSON Pascal's Triangle Properties Explore the fascinating patterns and relationships hidden within Pascal's Triangle Pascal's Triangle is built using a beautiful recursive relationship. Each entry is the sum of the two entries directly above it. At the edges, we always place 1. Each row has remarkable properties. The sum of all entries in row n equals 2^n. Even more interesting, if we alternate signs (adding and subtracting entries), the sum equals 0 for all rows except the first. Pascal's Triangle contains infinite sequences along its diagonals. The first diagonal is all 1s. The second diagonal gives the natural numbers. The third diagonal produces triangular numbers (1, 3, 6, 10, 15...), and beyond that lie tetrahedral numbers and higher-dimensional figurate numbers. Here's a delightful surprise: if you read each row as a single number (concatenating digits), the first few rows give powers of 11! Row 0 is 1 = 11^0, row 1 is 11 = 11^1, row 2 is 121 = 11^2, row 3 is 1331 = 11^3. This works until carries become necessary. The famous Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) hides within Pascal's Triangle along "shallow diagonals." If you sum entries along diagonal paths that move right-and-down, you'll discover each Fibonacci number emerging from the triangle.

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