1+1+2+1+2+3+1+2+3+4...n
turn into
n*(n+1)/2?
1 = 1
1+2 = 3
1+2+3 = 6
^ Solving for n = 3, we need to calculate the number of dots in the triangle right above this sentence.
To get this triangle, we first need to make a square. This square below is n x (n+1), or 3 x 4.
This triangle has an area of 3 * 4 or 12. It also contains two "1+2+3" triangles.
Dividing this square by two gets the triangle that equals 6, which is a 1+2+3 triangle.
With n = 4, you do the same thing.
When you divide this 5 x 4 square by 2, you get a "1+2+3+4" triangle.
Therefore, n*(n+1) is the square which is made up of two triangles that have a sum of 1+2+...n, and when you divide it by two, you get one triangle with the sum from 1 -> n.
