## Sequences and Series

Patterns of numbers separated by commas are called sequences e.g. $2,4,6,8,10{\ldots }$ and $1,3,5,7,9{\ldots }$

A sequence of terms can be defined by a rule for the $n^{\rm th}$ term e.g. $u_{n} = 2n + 3$

This would generate the sequence $u_{1} = 5$, $u_{2} = 7$, $u_{3} = 9$, $u_{4} = 11{\ldots }$, written as $5,7,9,11{\ldots }$

An inductive definition for a sequence is given by stating the first term and a rule showing how to get to the next term from the previous one. For example, the definition $u_{1} = 1, u_{n+1} = 2u_{n} + 3$ would generate this sequence $1,5,13,29{\ldots }$

### Arithmetic Sequences

$a, a+d, a+2d \ldots $

These are sequences which have a common difference e.g. $4,6,8,10,12{\ldots }$

In this case the common difference is 2.

The $n^{\rm th}$ term of an arithmetic sequence $u_{1}, u_{2}, u_{3}{\ldots }$ is given by $u_{n} = a + (n-1) d$ where

\begin{eqnarray*} a & =& {\hbox{first term}} \\ n & =& {\hbox{number of terms}} \\ d & =& {\hbox{common difference}} \\ \end{eqnarray*}

### Arithmetic Series

When the terms of an arithmetic sequence are added together they form an arithmetic series, e.g. $ 2 + 3 + 4 + 5 + 6 + {\ldots }$

The sum to n terms of an arithmetic series is:

\[ S_ n = {1\over 2}n(a+\ell ) = {1\over 2} n\left\{ 2a+(n-1)d\right\} \]

where

\begin{eqnarray*} a & =& {\hbox{first term}} \\ n & =& {\hbox{number of terms}} \\ d & =& {\hbox{common difference}} \\ \ell & =& {\hbox{last term}} \\ \end{eqnarray*}

### Geometric Sequences

$a, ar, ar^2 \ldots $

These are sequences which have a common ratio e.g. $2,6,18,54,162{\ldots }$

In this case the common ratio is 3.

To find the $n^{th}$ term use the formula

$n^{th} \hbox{term} = ar^{n-1}$

where

\begin{eqnarray*} a & =& \hbox{first term} \\ r & =& \hbox{common ratio} \\ n & =& \hbox{number of terms} \\ \end{eqnarray*}

### Geometric Series

The sum of the first $n$ terms of a geometric series is:

\[ S_ n = {a(1-r^ n)\over 1-r} \]

where

\begin{eqnarray*} a & =& \hbox{first term} \\ r & =& \hbox{common ratio} \\ \end{eqnarray*}

### The sum to infinity of a geometric series

\[ S_\infty = {a\over 1-r} \quad -1<r<1 \]