This example shows how to derive an expression for the sum of counting numbers from to using a geometric interpretation of the series.

Start with the series of counting numbers up to some number .

Visualize the base cases.

Expression Sum Observe that the summation can be modeled as the area of rectangles whose area represents a counting number. This is illustrated below.

The first three cases where the sum is represented as area are shown below.

Observe that the geometric area corresponding to terms can be represented as 1) the shape of a triangle whose base and height are equal to and 2) small triangles whose area is equal to .

The area of the triangle plus small triangles is given below.

Factor out to simplify the expression:

Because the area of this shape is equal to the series of counting numbers we have found an expression for the series.