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A triangular number is a figurate number: a number that can be represented by a regular geometric arrangement of equally spaced points.

Triangular numbers can be thought of as the numbers of dots you need to make a triangle:

 

The triangular numbers are found in the third diagonal of Pascal's triangle:
 
 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

   Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

   Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

   Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

   Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

   Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

   Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

   Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

   Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

   Row 8

References

  1. Ask Dr. Math Archives: What is a triangular number?   Why is 1 a triangular number?
  2. Ask Dr. Math FAQ: Pascal's Triangle
  3. Eric's Treasure Trove of Mathematics: Triangular Number


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Math Forum * * * * 2 May 1998