The capital letter Sigma Σ represents the summation operation in mathematics, often called sum for short. A sum has three parts: an initial value, an end value, and the expression being summed. The summation starts at the initial value and iterates, adding one to the value for each iteration, stopping when the value reaches the end value.
For example, the summation from i = 1 to 3 of the expression shown below is equal to one plus two plus three. Since the expression being summed is the variable i, then i starts at one, iterates to two, iterates to three, and then stops there since it is the ending value:
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.
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:
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.