Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers



Summation and Product Notation

Summation Notation
Product Notation

In this content note we discuss and illustrate compact mathematical notation to express certain types of sums and products.

Summation Notation

At times when we add, there is a pattern by which we can express the addends. For instance, in the sum

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

the smallest addend is 1, each successive addend is one larger than the one before it, and the largest addend is 10. Likewise, in the sum

2 + 6 + 10 + 14 + 18

the smallest addend is 2, each successive addend is 4 larger than the previous, and the largest addend is 18. See whether you can detect and describe the addend patterns in the following sums.

(1)
3 + 6 + 9 + 12 + 15 + 18 + 21
(2)
5 + 10 + 20 + 40 + 80 + 160
(3)
2a + 4a + 6a + 8a + 10a + 12a + 14a
(4)
1 + 4 + 9 + 16 + 25 + 36 + . . .
(5)
3 + 6 + 9 + 12 + 15 + 18 + 21 + . . .
(6)
10 - 16 + 22 - 28 + 34 - . . .

Summation notation provides for us a compact way to represent the addends in sums such as these. For instance, here is the summation notation to represent the sum of the first 10 positive integers, the first sum described on this page.

The annotated symbolism shown below identifies important elements used in summation notation (also called sigma notation).

To expand this summation notation, that is, to determine the set of addends that we are to sum, we replace any occurance of the dummy variable in the addend representation with the lower limit of the index variable. We evaluate the resulting expression. This is our first addend. We repeat this process with the next value of the index variable, using that specific value for the index variable in the addend representation and simplifying as desired or necessary. The replace and simplify process continues until the last index value to be used is the upper limit of summation.

Determine the expansion of this summation notation:

Each addend in the sum will be the square of an index value. The index values begin with 3 and increase by 1 until reaching 7. Thus, we have the index values 3, 4, 5, 6, and 7, and the squares of those are 9, 16, 25, 36, and 49. The summation notation above, therefore, represents the sum 9 + 16 + 25 + 36 + 49.

In some cases we may not identify the upper limit of summation with a specific value, instead usingf a variable. Here's an example.

The lower limit of summation is 0 and the upper limit is n. Each addend in the sum is found by multiplying the index value by 3 and then adding 1 to that. When j=0, the addend is (3)(0)+1=1. When j=1, the addend is (3)(1)+1=4. When we reach the upper limit, the addend is (3)(n)+1. Because we do not know the specific value for n, we use an elipsis (. . .) to signal that the addend pattern continues. Here's the expansion of this summation notation.

1 + 4 + 7 + ... + (3n+1)

We may also create sums with an infinite number of addends. In this situation, the upper limit of summation is infinity. Here's an example.

When k=1, the addend is (1+1)^3=8, when k=2 the addend is (2+1)^3=27, and so on. There is no last addend, because the upper limit of summation is infinity, indicating we simply continue to create addends following the pattern shown. Here's the expansion for this infinite summation.

Product Notation

Once you've learned how to use summation notation to express patterns in sums, product notation has many similar elements that make it straightforward to learn to use. The only difference is that we use product notation to express patterns in products, that is, when the factors in a product can be represented by some pattern. Instead of the Greek letter sigma, we use the Greek letter pi. Here is product notation to represent the product of the first several squares:


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