Triangular Numbers
Numbers that form equilateral triangles when arranged as dots: where geometry meets arithmetic
A triangular number counts the objects that can form an equilateral triangle. The sequence begins 1, 3, 6, 10, 15, 21, 28... where each term adds the next natural number: T(1) = 1, T(2) = 1+2 = 3, T(3) = 1+2+3 = 6, and in general T(n) = n(n+1)/2. First studied by the Pythagoreans over 2,500 years ago, triangular numbers connect geometry, algebra, and number theory in elegant ways.
Visual Representation
Triangular numbers get their name from the way they can be arranged as dots forming equilateral triangles. Each row adds one more dot than the previous row:
*
*
* *
*
* *
* * *
*
* *
* * *
* * * *
*
* *
* * *
* * * *
* * * * *
The Formula: Gauss's Insight
The n-th triangular number is given by the formula T(n) = n(n+1)/2. This elegant formula was famously derived by the young Carl Friedrich Gauss when his teacher asked the class to sum the numbers from 1 to 100.
Gauss's trick:
Pair the numbers from opposite ends of the sequence. Each pair sums to the same value:
50 pairs x 101 = 5,050. Therefore T(100) = 5,050.
In general: T(n) = n(n+1)/2. This works because there are n/2 pairs, each summing to (n+1).
Properties of Triangular Numbers
- The sum of two consecutive triangular numbers is always a perfect square: T(n) + T(n+1) = (n+1)2.
- A number n is triangular if and only if 8n + 1 is a perfect square.
- The difference between consecutive triangular numbers increases by 1 each time: T(n+1) - T(n) = n+1.
- Every triangular number is a binomial coefficient: T(n) = C(n+1, 2), appearing in row 2 of Pascal's triangle.
Connections to Other Number Types
Triangular numbers have deep connections to many other sequences in mathematics:
- Some triangular numbers are also perfect squares (called square triangular numbers): 1, 36, 1,225, 41,616...
- Every perfect number is a triangular number. For example, 6 = T(3) and 28 = T(7).
- Hexagonal numbers are a subset of triangular numbers: every hexagonal number H(n) = T(2n-1).
Table of the First 20 Triangular Numbers
Here is a compact reference table showing T(n) for n = 1 to 20, along with their values:
The First 50 Triangular Numbers
Click on any triangular number to see its complete mathematical analysis.
Did You Know?
- Gauss reportedly solved the sum 1 + 2 + ... + 100 = 5,050 at the age of 7, astonishing his teacher.
- Every positive integer can be represented as the sum of at most three triangular numbers (Gauss's eureka theorem, 1796).
- The numbers of balls in a standard bowling arrangement (10), billiards rack (15), and snooker triangle (15) are all triangular numbers.
- The 12th triangular number is 78, which equals the total number of cards dealt from a standard 52-card deck in a 13-round game of bridge.
- Triangular numbers appear in Pascal's triangle as the diagonal sequence: 1, 3, 6, 10, 15, 21...
Preguntas Frecuentes
What is a triangular number?
A triangular number is the sum of the first n natural numbers: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, and so on. The name comes from the fact that these numbers of objects can be arranged in equilateral triangle patterns.
What is the formula for the n-th triangular number?
The n-th triangular number is T(n) = n(n+1)/2. For example, the 10th triangular number is T(10) = 10 x 11 / 2 = 55. This formula was popularized by the legendary mathematician Carl Friedrich Gauss.
How do I check if a number is triangular?
A number m is triangular if and only if 8m + 1 is a perfect square. For example, is 21 triangular? 8 x 21 + 1 = 169 = 132. Since 169 is a perfect square, yes, 21 is the 6th triangular number.