Triangular Number Sequence

This is the Triangular Number Sequence:

This sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

A Rule

We can make a "Rule" so we can calculate any triangular number.

First, rearrange the dots (and give each pattern a number n), like this:

Then double the number of dots, and form them into a rectangle:

The rectangles are n high and n+1 wide (and remember we doubled the dots), and x_n is how many dots (the value of the Triangular Number n):

Rule: x_n = n(n+1)/2