Triangular numbers

Polygonal numbers are really just the number of vertexes in a figure formed by a certain polygon. The first number in any group of polygonal numbers is always 1, or a point. The second number is equal to the number of vertexes of the polygon. The third polygonal number is made by extending two of the sides of the polygon from the second polygonal number, completing the larger polygon and placing vertexes and other points where necessary. The third polygonal number is found by adding all the vertexes and points in the resulting figure ...

The triangular number is a polygonal number : a number that can be represented by a regular geometric arrangement of equally spaced points. As the name suggests triangular numbers can be visualised as a triangle of points :

The triangular numbers can be found in the third diagonal of Pascal`s triangle, starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the forth is 10, and so on.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1

Traingular numbers can be calculated by adding up consectuive numbers. For example, the seventh triangular number is equal

which comes to 28

Numbers that are Added

The Sum of These
(the new pattern)

1 1 + 2

3 1 + 2 + 3

6 1 + 2 + 3 + 4

10 1 + 2 + 3 + 4 + 5

15 1 + 2 + 3 + 4 + 5 + 6

21 1 + 2 + 3 + 4 + 5 + 6 + 7

28 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

36 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

45 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

55

An interesting number sequence can be generated by adding up tringular numbers as shown in the table below :

Numbers that are Added

The Sum of These
(the new pattern)

1 1 + 3

4 1 + 3 + 6

10 1 + 3 + 6 + 10

20 1 + 3 + 6 + 10 + 15

35 1 + 3 + 6 + 10 + 15 + 21

56 1 + 3 + 6 + 10 + 15 +21 + 28

84 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36

120 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45

165 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55

220

The sum of these ... the new pattern shown in the right column ... is referenced as the sequence of tetrahedral numbers :

The triangular number is a figurate number of the form

where n= 2, 3, 4, 5, ...

As it is known, the natural number is a figurate number of the following form

where n= 1, 2, 3, 4, 5, ....

There is a connection with triangular numbers and natural numbers :

Now, the operator`s relation is obtained

or :

An infinite sum of the natural numbers as the negative triangular number can be calculated, as shown in the table below :

Infinite series of the natural numbers