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1: + x3: x x + + x x6: x x x x x x + + + x x x10: x x x x x x x x x x x x + + + + x x x x15: x x x x x x x x x x x x x x x x x x x x + + + + + x x x x x21: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + x x x x x x |
Since each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers.
The formula for the nth triangular number is ½n(n + 1) or (1 + 2 + 3 + ... + [n − 2] + [n − 1] + n).
It is the binomial coefficient
It can also be shown that for any n-dimensional simplexTopology Geometry Geometry In geometry, a simplex is an n dimensional figure, being the convex hull of a set of n + 1) affinely independent points in some Euclidean space of dimension n or higher i. a set of points such that no m plane contains more than with sides of length x, the formula
yields the number of points that make up the simplex. For example, a tetrahedronA tetrahedron (plural: tetrahedra is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids. The area A with sides of length 2 corresponds to the number (2)(2 + 1)(2 + 2)/6, or 4. The four points forming this configuration are the verticesIn geometry, a vertex ( Latin: whirl, whirlpool; plural vertices is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In graph theory, a graph describes a set of connections betw of the tetrahedron. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangled = 3 plus 1 triangled = 1 = 4.)
One of the most famous triangular numbers is 666666 Six hundred sixty-six in American English, Six hundred and sixty-six elsewhere) is the Number of the Beast in the Christian Bible, in the Book of Revelation. There is also a house music group named "666". For the year, see AD 666. 666 CardinalSix hund, also known as the Number of the BeastThe Mark of the Beast is mentioned in the Book of Revelation of the Christian Bible. References from the Bible Revelation 13:17 states: :". no one may buy or sell except one who has the mark or name of the beast, or the number of his name. The next verse. Every evenIn mathematics, any integer (whole number) is either even or odd . If it is a multiple of two, it is an even number otherwise, it is an odd number . Examples of even numbers are −4, 8, 0, and 70. Examples of odd numbers are −5, 1, and 71. perfect numberIn mathematics, a perfect number is an integer which is the sum of its proper positive divisors, excluding itself. Thus, 6 is a perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 6. The next perfect number is 28 1 + 2 + 4 + is triangular.
The sum of two consecutive triangular numbers is a square numberIn mathematics, a square number sometimes also called a perfect square is a positive integer that can be written as the square of some other integer. So for example, 9 is a square number since it can be written as 3×3. By convention, the first square numb. This can be shown mathematically thus: the sum of the nth and (n-1)th triangular numbers is {½n(n + 1)} + {½(n − 1)n}. This simplifies to (½n2 + ½n) + (½n2 − ½n), and thus to n2. Alternatively, it can be demonstrated diagrammatically, thus:
x + + + x x + + x x x + x x x x x + + + + x x + + + x x x + + x x x x + x x x x xIn each of the above examples, a square is formed from two interlocking triangles.
More generally, the difference between the nth m -gonal number and the nth (m+1)-gonal number is the (n-1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15.
Also, the square of a triangular number n is the same as the sum of the cubes of the integers 1 to n.
In base 10, the digital root of a triangular number is always 1, 3, 6 or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine.
6=3*2, 10=9+1, 15=3*5, 21=3*7, 28=9*3+1, ...Triangular numbers have all sorts of relations to other figurate numbers, including centered figurate numbers. Whenever a triangular number is divisible by 3, one third of it will be a pentagonal number. Every other triangular number is a hexagonal number. A centered hexagonal number is a triangular number multiplied by 6, plus 1. A centered square number is a triangular number multiplied by 4, plus 1.