Triangular Numbers
Numbers that form perfect triangles with dots
Triangular numbers are those that can be represented as an equilateral triangle of dots. The nth triangular number is obtained by adding the first n natural numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55... The formula is T(n) = n(n+1)/2.
Visualization
The first triangular numbers can be represented graphically as triangles of dots:
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The Formula: T(n) = n(n+1)/2
There is a famous anecdote about the mathematician Carl Friedrich Gauss. When he was a child, his teacher asked the class to add all the numbers from 1 to 100, expecting to keep them busy for quite a while. However, young Gauss found the answer in seconds.
Gauss's trick:
He noticed that the numbers could be paired from both ends:
There are 50 pairs, each summing to 101, so the total is 50 × 101 = 5,050.
Generalized: T(n) = n(n+1)/2
Properties of Triangular Numbers
- Gauss's eureka theorem: Every natural number is the sum of at most 3 triangular numbers.
- Perfect squares: T(n) + T(n−1) = n². Two consecutive triangular numbers always sum to a perfect square.
- Relationship with squares: 8·T(n) + 1 is always a perfect square. For example: 8×6 + 1 = 49 = 7².
- Sum of triangulars: The sum of the first n triangular numbers is n(n+1)(n+2)/6, which are the tetrahedral numbers.
Relationship with Other Numbers
Triangular numbers have fascinating connections with other types of numbers:
- Triangular and square: Some triangular numbers are also perfect squares: 1, 36, 1,225, 41,616...
- Palindromic triangulars: Some triangular numbers are also palindromes, such as 1, 3, 6, 55, 66, 171, 595...
- Pascal's triangle: Triangular numbers appear on the third diagonal of Pascal's triangle (the binomial coefficients C(n,2)).
The First 20 Triangular Numbers
Table with the index n and its corresponding triangular number T(n):
The First 50 Triangular Numbers
Click any triangular number to see its full analysis:
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