# Triangular number

The **triangular numbers** are the numbers which are the sum of the first natural numbers from to .

## Definition

The triangular number is the sum of all natural numbers from one to n. That is, the triangle number is .

For example, the first few triangular numbers can be calculated by adding 1, 1+2, 1+2+3, ... etc. giving the first few triangular numbers to be .

A rather simple recursive definition can be found by noting that .

They are called triangular because you can make a triangle out of dots, and the number of dots will be a triangular number:

## Formula

Using the sum of an arithmetic series formula, a formula can be calculated for :

The formula for finding the triangular number can be written as .

It can also be expressed as the sum of the row in Pascal's Triangle and all the rows above it. Keep in mind that the triangle starts at Row 0.

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