Properties of the summation symbol

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:

$\begin{aligned}\sum_{k=1}^{10}2k&=2\cdot \sum_{k=1}^{10}k\\[0.25cm]&=2\cdot(1+2+3+4+5+6+7+8+9+10)\\[0.25cm]&=2\cdot 55\\[0.25cm]&=110\end{aligned}$