Elementary combinatorics: Summation and product symbol
Properties of the summation symbol
A summation can be calculated in different ways, for example through a different arrangement of the terms. The following properties of the summation symbol come into play here.
Sum rule \[\sum_{i=m}^{n}(a_i+b_i)=\sum_{i=m}^{n}a_i+\sum_{i=m}^{n}b_i\]
Constant factor rule \[\sum_{i=m}^{n}c\cdot a_i=c\cdot \sum_{i=m}^{n}a_i\] for some constant \(c\).
Summation of a constant \[\begin{aligned}\sum_{i=m}^{n}c&=c\cdot \sum_{i=m}^{n}1\\[0.25cm] &=c\cdot(n-m+1)\end{aligned}\] for some constant \(c\).
Change of boundaries and index \[\sum_{i=m}^{n} a_i= \sum_{j=m+r}^{n+r}a_{j-r}\] for some integer \(r\), whee we have replaced the summation index \(i\) by \(j\) via \(j=i+r\).
Commutativity rule \[\sum_{i=1}^{n}\sum_{j=1}^{m} a_{ij}=\sum_{j=1}^{m}\sum_{i=1}^{n} a_{ij}\] where we have used a double index \(ij\) here.
Example 1 You can calculate \[\sum_{k=1}^{10}2k\] in at least two ways: by adding up all the terms and by using the constant factor rule. \[\begin{aligned}\sum_{k=1}^{10}2k&= 2+4+6+8+10+12+14+16+18+20\\[0.25cm]&=110\end{aligned}\] But also using. the constant factor rule: \[\begin{aligned}\sum_{k=1}^{10}2k&=2\cdot \sum_{k=1}^{10}k\\[0.25cm]&=2\cdot(1+2+3+4+5+6+7+8+9+10)\\[0.25cm]&=2\cdot 55\\[0.25cm]&=110\end{aligned}\]
