Product symbol

Elementary combinatorics: Summation and product symbol

Product symbol

A sum with many terms can be written in a concise way using the summation symbol. Similarly, a product with many factors can be written in a short way using the product symbol $\prod$ , which comes from the Greek capital letter pi.

$\prod_{i=m}^{n}a_i=a_m\cdot a_{m+1}\cdots a_{n-1}\cdot a_n$ This refers to the product of mathematical objects where the multiplication index $i$ is increased by one unit at a time, starting at $i=m$ and ending at $i=n$ .

Example 1

$\displaystyle\prod_{n=1}^4n= 1\times 2\times 3\times 4 = 24$

Example 2

$\displaystyle\prod_{k=0}^3(x-k)= x(x-1)(x-2)(x-3)$

The geometric mean $\mu$ of $n$ numbers $a_1, a_2,\ldots, a_n$ is defined as the $n$ -th root of the product of these $n$ numbers. In formula form:

$\mu=\left(\prod_{i=1}^{n}a_i\right)^{\frac{1}{n}}\!\!=\sqrt[n]{a_1a_2\cdots a_n}$

The properties of the product symbol are similar to those of the summation symbol.

Product rule

$\prod_{i=m}^{n} (a_i\cdot b_i)=\prod_{i=m}^{n} a_i\cdot \prod_{i=m}^{n}b_i$

Constant factor rule

$\prod_{i=m}^{n} (c\cdot a_i)=c^{m-n+1}\prod_{i=m}^{n} a_i$ for some constant $c$ .

Change of bounds and index

$\prod_{i=m}^{n} a_i= \sum_{j=m+r}^{n+r}a_{j-r}$ for some integer $r$ , where we have replaced the product index $i$ by $j$ via $j=i+r$ .

Commutativity rule

$\prod_{i=1}^{n}\prod_{j=1}^{m} a_{ij}=\prod_{j=1}^{m}\prod_{i=1}^{n} a_{ij}$ where we have used a double index $ij$ here.