5.1 Approximating areas  (Page 2/17)

Rule: properties of sigma notation

Let $a_1,a_2\text{,…,}{a}_n$ and $b_1,b_2\text{,…,}{b}_n$ represent two sequences of terms and let c be a constant. The following properties hold for all positive integers n and for integers m , with $1\le m\le n.$

1. 
   
   ${\displaystyle \sum _{i=1}^{n}c}=nc$
2. 
   
   ${\displaystyle \sum _{i=1}^{n}c{a}_i}=c{\displaystyle \sum _{i=1}^n{a}_i}$
3. 
   
   ${\displaystyle \sum _{i=1}^n\left(a_i+b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}+{\displaystyle \sum _{i=1}^n{b}_i}$
4. 
   
   ${\displaystyle \sum _{i=1}^n\left(a_i-b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}-{\displaystyle \sum _{i=1}^n{b}_i}$
5. 
   
   ${\displaystyle \sum _{i=1}^n{a}_i}={\displaystyle \sum _{i=1}^m{a}_i}+{\displaystyle \sum _{i=m+1}^n{a}_i}$

Rule: sums and powers of integers

1. The sum of n integers is given by
   
   ${\displaystyle \sum _{i=1}^{n}i}=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}.$
2. The sum of consecutive integers squared is given by
   
   ${\displaystyle \sum _{i=1}^n{i}^2}=1^2+2^2+\text{⋯}+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}.$
3. The sum of consecutive integers cubed is given by
   
   ${\displaystyle \sum _{i=1}^n{i}^3}=1^3+2^3+\text{⋯}+n^3=\frac{n^2{\left(n+1\right)}^2}{4}.$