Sum of Counting Numbers
To calculate the summation of the counting numbers 1 + 2 + 3 + ... + n the sequence can be visualized geometrically and solved for by finding the area of the shape formed.
Steps

As with many summation problems a good first step is to visualize the base cases for the summation. Then the sequence can be investigated to see if there is a common pattern that can be reprsented using the variable .
Expression Sum 
We can visualize this summation as the sum of rectangles of width . This transforms the problem into a geometry problem where we want to find the area of a shape.
The first three expressions are represented as the shapes below.
We can observe that the general form of the expression can be represented 1) by the area of a triangle with a base of and a height of and 2) by triangles with an area of .

This observation gives us the expression in terms of that represents the summation of the counting numbers.

It is always a good idea to check our work and shown below is the for our expression holds true.