Sums of Integers, Squares & Cubes (Edexcel A Level Further Maths: Core Pure): Revision Note

Sums of integers, squares & cubes

How can we use sigma notation?

• When writing the sum of a series you can use sigma notation

  • The series $u_1+u_2+u_3+...+u_{n-1}+u_n$ can instead be written as $\sum _{r=1}^{r=n}{u}_r$ or $\sum _{r=1}^n{u}_r$

  • This means “the sum of all the terms from $u_1$ to $u_n$for the sequence described by $u_r$"

  • $\sum _{r=1}^{3}2r+3$ would mean $\left(2\right(1)+3)+\left(2\right(2)+3)+\left(2\right(3)+3)=21$

• Using the following relations, summations can be grouped together (or ungrouped) to make some calculations easier:

$\sum _{r=p}^{q}a\mathrm{f}\left(r\right)+b=\sum _{r=p}^{q}a\mathrm{f}\left(r\right)+\sum _{r=p}^{q}b=\underset{r=p}{\overset{q}{a\sum }}\mathrm{f}\left(r\right)+\sum _{r=p}^{q}b$

$\sum _{r=p}^{q}a\mathrm{f}\left(r\right)+b\mathrm{g}\left(r\right)=\sum _{r=p}^{q}a\mathrm{f}\left(r\right)+\sum _{r=p}^{q}b\mathrm{g}\left(r\right)=\underset{r=p}{\overset{q}{a\sum }}\mathrm{f}\left(r\right)+b\sum _{r=p}^q\mathrm{g}\left(r\right)$

• a and b are constants, and $\mathrm{f}\left(r\right)$ and $\mathrm{g}\left(r\right)$ are a functions of r

• Note that the top and bottom summation limits ($p$ and $q$) are the same for all the sums

  • This is important – if the top and bottom limits don’t all match then the relation is no longer valid!

• This can be very useful when $\mathrm{f}\left(r\right)$ or $\mathrm{g}\left(r\right)=r,r^2$ or $r^3$, as and  (or $r^2$ or $r^3$) are straightforward to find using formulae

• A useful result to remember is that $\sum _{r=1}^{n}a=a+a+a+...\left(n\mathrm{times}\right)=n\times a$

What are the formulae for finding sums of integers, squares, and cubes?

• There are several useful formulae for summing integers, square numbers, and cube numbers

• The sum of the first $n$ natural numbers is given by 

  • $\sum _{r=1}^{n}r=\frac{1}{2}n(n+1)$

    • This formula is not given in the formula book

• The sum of the first $n$ square numbers is given by

  • $\sum _{r=1}^n{r}^2=\frac{1}{6}n(n+1)(2n+1)$ 

    • This formula is given in the formula book

• The sum of the first $n$ cube numbers is given by

  • $\sum _{r=1}^n{r}^3=\frac{1}{4}{n}^2{(n+1)}^2$ 

    • This formula is given in the formula book

    • Notice that this is equal to the formula for the sum of the first $n$ natural numbers, squared

• Using the relations given above, a more complicated summation can often be broken down into sums of constants, natural numbers, squares, and cubes

  • For example,