Concept 18 of 19Advanced
Watch on YouTubeVideoPascal's triangle
A triangular array of numbers. Start with 1 at top. Each number = sum of the two above.
Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4:1 4 6 4 1 Row 5: 1 5 10 10 5 1 Row 6:1 6 15 20 15 6 1
Hidden patterns:
- Second diagonal: 1, 2, 3, 4, 5 (natural numbers)
- Third diagonal: 1, 3, 6, 10, 15 (triangular numbers)
- Fourth diagonal: 1, 4, 10, 20, 35 (tetrahedral numbers!)
- Sum of each row: 1, 2, 4, 8, 16, 32 (powers of 2)
- Shallow diagonals sum to Fibonacci numbers
Example
Row 4 sum = 1+4+6+4+1 = 16 = 2⁴.
Fourth diagonal gives 1, 4, 10, 20, 35 — exactly our tetrahedral numbers.
Fourth diagonal gives 1, 4, 10, 20, 35 — exactly our tetrahedral numbers.
💡 Tip:Pascal's triangle is a 'cheat sheet' connecting counting, triangulars, tetrahedrals, powers of 2, and Fibonacci.
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5 questions to check what you just read.
0 / 5
- Q1.Row 4 of Pascal's triangle: 1, 4, 6, 4, 1. Sum of the row:
- Q2.3rd entry in row 5 (1, 5, 10, 10, 5, 1)?
- Q3.The 4th diagonal of Pascal's triangle gives:
- Q4.Middle entry of row 6 (1, 6, 15, 20, 15, 6, 1)?
- Q5.The sum of entries in row n of Pascal's triangle: