Hence, the

triangular number can reduce the information distortion and information deviation in the process of decision-making.

Hence a

triangular number (a1; aM; a2) can be written in the form of a trapezoidal number, i.

Proof: (i) ([right arrow]) Assume T is a

triangular number.

2]

Triangular numbers [ILLUSTRATION OMITTED] 1 1 3 1 + 2 6 1 + 2 + 3 10 1 + 2 + 3 + 4 15 1 + 2 + 3 + 4 + 5 .

It is a reasonably well-known fact that the sum of consecutive cubes is the square of the corresponding

triangular number, that is

Were they able to articulate the relationship between a

triangular number and its position in the sequence?

Proof: T is a perfect number [&&] T is a

triangular number.

For example, students might try to determine the maximum number of iterations, or repetitions, needed to reach the marching group for a given

triangular number of seeds distributed among consecutive holes.

Note that the first

triangular number is associated with the fictitious triangle of side length zero units.

Each subsequent number in the

triangular number sequence is created by adding a row to the bottom of the triangle.

while the other Smarandache sequences of

triangular numbers only show, among the first 1000 terms, the trival

triangular number 1:

Using the sum of an arithmetic progression with common difference 1, it can be shown that the nth

triangular number is given by