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