Calculate Sum of Counting Numbers

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


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

    Visualize the base cases.

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

    The geometric sum of counting numbers

    The total area (sum) for the first three cases is shown below.

    The area of the base cases

  3. Observe that the general form of the expression can be represented as the shape of a triangle plus little triangles whose area is equal to . The area of the big triangle is equal to where the base and height are equal to .

    The area of the base cases

    The expression for the area is given below.

    Factoring out the from both expressions simplifies the expression:

  4. Finally, because the area of this shape is equal to the series of counting numbers we have found an expression for the series.