# Sequences and Series

## Contents

## Number Patterns[edit]

An important skill in mathematics is to be able to:

**recognise**patterns in sets of numbers,**describe**the patterns in words, and**continue**the pattern

A list of numbers where there is a pattern is called a **number sequence**. The members (numbers) of a sequence are said to be its **terms**.

### Example[edit]

The above is a type of number sequence. The first term is , the second is , etc. The rule of the sequence is that "*the sequence starts at 3 and each term is 4 more than the previous term.*"

## Arithmetic Sequences[edit]

An *arithmetic sequence* is a sequence in which each term differs from the previous by the same fixed number:

is arithmetic as etc

### Algebraic Definition[edit]

Within an arithmetic sequence, the -th term is defined as follows:

Where is defined as:

Here, the notation is as follows:

is the first term of the sequence.

is the number of terms in the sequence.

is the common difference between terms in an *arithmetic* sequence.

### Example[edit]

Given the sequence , the values of the notation are as follows:

And

Therefore

Thus we can determine any value within a sequence:

## Arithmetic Series[edit]

An *arithmetic series* is the addition of successive terms of an arithmetic sequence.

### Sum of an Arithmetic Series[edit]

Recall that if the first term is and the common difference is , then the terms are:

Suppose that is the last or final term of an arithmetic series. Then, where is the sum of the arithmetic series:

One can see that there there in fact terms that look identical, thus:

## Geometric Sequences[edit]

A sequence is *geometric* if each term can be obtained from the previous one by multiplying by the same non-0 constant.

is geometric as and and .

Notice that

i.e., each term divided by the previous one is a non-0 constant.

### Algebraic definition[edit]

is **geometric** for all positive integers , where is a **constant** (the **common ratio**).

### The 'Geometric' Mean[edit]

If are any consecutive terms of a geometric sequence, then

{equating common ratios}

Therefore

and so where is the **geometric mean** of .

### The General Term[edit]

Suppose the first term of a geometric sequence is and the common ratio is .

Then therefore etc.

Thus

is the first term of the sequence.

is the general term.

is the common ratio between terms in an *geometric* sequence.

## Geometric Series[edit]

## Compound Interest[edit]

Compound interest is the interest earned on top of the original investment. The interest is *added* to the amount. Thus the investment grows by a large amount each time period.

- Consider the following

You invest $1000 into a bank. You leave the money in the back for 3 years. You are paid an interest rate of 10% per annum (p.a.). The interest is added to your investment each year.

An interest rate of 10% p.a. is paid, *increasing the value* of your investment yearly.

Your percentage increase each year is 10%, i.e.,

So 110% of the value at the start of the year, which corresponds to a *multiplier* of 1.1.

After one year your investment is worth

After two years it is worth |
After three years it is worth |

- Note

The initial investment | ||

The amount after 2 year | ||

The amount after 3 years | ||

The amount after 4 years | ||

The amount after years |

In general, is used for compound growth, where

is the initial investment

is the growth multiplier

is the number of years

is the amount after years