Sum of Counting Numbers

To calculate the summation of the counting numbers from to in the form of: the sequence can be visualized geometrically and solved for by finding the area of the shape formed.


  1. 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 . The first four cases are represented in the table below:

    Expression Sum
  2. 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.

    Sum of 1 + 2 + 3 + ... + n geometric

    The first three expressions are represented as the shapes below.

    Geometric expressions for sum of counting numbers

    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 .

    Geometric Form of Sum of Counting Numbers
  3. This observation gives us the expression in terms of that represents the summation of the counting numbers.

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