# Sigma notation

Level: A Level Maths

Sigma notation is used to define the sum of a series, in a concise way.

## Series of natural numbers

$1+2+3+\cdots +n$ Can be written as:

$\sum _{r\mathop {=} 1}^{n}r$ This can be extended for powers of $r$ . For example the sum of the cubes of the first n natural numbers can be written like this:

$\sum _{r\mathop {=} 1}^{n}r^{3}=1^{3}+2^{3}+3^{3}+\cdots +n^{3}$ ## Arithmetic progression

A general arithmetic progression takes the form:

$a,a+d,a+2d,a+3d,\cdots$ The rth term is

$a+(r-1)d$ The sum of the first $n$ terms can be written as:

$\sum _{r\mathop {=} 1}^{n}{a+(r-1)d}$ We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from $r=0$ , we only count up to $r=n-1$ . We can rewrite the sum like this:

$\sum _{r\mathop {=} 0}^{n-1}{a+rd}$ ## Geometric progression

A general geometric progression takes the form:

$a,ar,a{r^{2}},a{r^{3}},\cdots$ The kth term is

$a{r^{k-1}}$ The sum of the first $n$ terms can be written as:

$\sum _{k\mathop {=} 1}^{n}a{r^{k-1}}$ We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from $r=0$ , we only count up to $r=n-1$ . We can rewrite the sum like this:

$\sum _{k\mathop {=} 0}^{n-1}a{r^{k}}$ ## General case

For function $F(r)$ the sum

$F(1)+F(2)+F(3)\cdots +F(n)$ can be written as:

$\sum _{r\mathop {=} 1}^{n}F(r)$ And the sum

$F(m)+F(m+1)+F(m+2)\cdots +F(n-1)+F(n)$ can be written as:

$\sum _{r\mathop {=} m}^{n}F(r)$ 