# 3.1 Sigma notation

One way of writing long sums (commonly used at GCSE), is to use an elipsis (the $...$ symbol). For example, if we were to write the sum of all the positive integers from $1$ to $21$ we could write it out in full as

 $\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12$ (3.1) $\displaystyle+13+14+15+16+17+18+19+20+21$

The other way we could write it is as

$1+2+3+...+21$ (3.2)

Here we’ve used $...$ to stand for all the terms between 2 and 21. This notation works, but there’s another way we could write this sum; using sigma notation! In this case, we could write this as

$\sum_{i=1}^{21}i$ (3.3)

Sigma notation is not as bad as it looks! All this means (when read aloud) is \saythe sum of all the values of $i$ where $i$ starts at $1$ and ends at $21$ (inclusive11 1 i.e. we include $1$ and $21$).

In general there are three main parts to sigma notation - the place where we start counting from, the place where we finish counting and the "variable of indexation". In a more general case, we would have something of the form

$\sum_{i=0}^{n}\left[\text{some expression depending on i}\right]$ (3.4)

This would mean that we start at $i=0$ and find the value of whatever the expression depending on $i$ is. Say, for the sake of example, it happened to be22 2 This is just an example - the expression could be anything! $3i^{2}$. In this case, we would have $3\cdot 0^{2}=0$. We would then add this to the value of the expression at $1$ ($3\cdot 1^{2}$), at $2$ ($3\cdot 2^{2}$), at $3$ ($3\cdot 3^{2}$), and so on (all the way to $n$ - $3\cdot n^{2}$).