# Sequences and Series

## Number Patterns

An important skill in mathematics is to be able to:

• recognise patters in sets of numbers,
• describe the patters in words, and
• continue the pattern

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

### Example

${\displaystyle 3,7,11,15,...}$

The above is a type of number sequence. The first term is ${\displaystyle 3}$, the second is ${\displaystyle 7}$, 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

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

${\displaystyle 2,5,8,11,14,...}$ is arithmetic as ${\displaystyle 5-2=8-5=11-8=14-11}$ etc

### Algebraic Definition

Within an arithmetic sequence, the ${\displaystyle n}$th term is defined as follows:

${\displaystyle u_{n}=u_{1}+(n-1)d}$

Where ${\displaystyle d}$ is defined as:

${\displaystyle d=u_{n+1}-u_{n}}$

Here, the notation is as follows:

 ${\displaystyle u_{1}}$ is the first term of the sequence. ${\displaystyle n}$ is the number of terms in the sequence. ${\displaystyle d}$ is the common difference between terms in an arithmetic sequence.

### Example

Given the sequence ${\displaystyle 1,3,5,7...n}$, the values of the notation are as follows:

${\displaystyle d=u_{n+1}-u_{n}}$

${\displaystyle d=u_{2}-u_{1}=3-1}$

${\displaystyle d=2}$

And

${\displaystyle u_{1}=1}$

Therefore

${\displaystyle u_{n}=u_{1}+(n-1)d}$

${\displaystyle u_{n}=1+(n-1)2}$

${\displaystyle u_{n}=1+2n-2}$

${\displaystyle u_{n}=2n-1}$

Thus we can determine any value within a sequence:

${\displaystyle u_{5}=2(5)-1=10-1=9}$

## Arithmetic Series

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

${\displaystyle 21+23+27+...+49}$

### Sum of an Arithmetic Series

Recall that if the first term is ${\displaystyle u_{1}}$ and the common difference is ${\displaystyle d}$, then the terms are:

${\displaystyle u_{1},u_{1}+d,u_{1}+2d,u_{1}+3d,...}$

Suppose that ${\displaystyle u_{n}}$ is the last or final term of an arithmetic series. Then, where ${\displaystyle S_{n}}$ is the sum of the arithmetic series:

${\displaystyle S_{n}=u_{1}+(u_{1}+d)+(u_{1}+2d)+...+(u_{n}-2d)+(u_{n}-d)+u_{n}}$

However, if one were to reverse the series, like so:

${\displaystyle S_{n}=u_{n}+(u_{n}-d)+(u_{n}-2d)+...+(u_{1}+2d)+(u_{1}+d)+u_{1}}$

Then add the two sequences together:

${\displaystyle 2S_{n}=(u_{1}+u_{n})+(u_{1}+u_{n})+(u_{1}+u_{n})+...+(u_{1}+u_{n})+(u_{1}+u_{n})+(u_{1}+u_{n})}$

One can see that there there in fact ${\displaystyle n}$ terms that look identical, thus:

${\displaystyle 2S_{n}=n(u_{1}+u_{n})}$

${\displaystyle S_{n}={\frac {n}{2}}(u_{1}+u_{n})}$

## Geometric Sequences

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

${\displaystyle 2,10,50,250,...}$ is geometric as ${\displaystyle 2\times 5=10}$ and ${\displaystyle 10\times 5=50}$ and ${\displaystyle 50\times 5=250}$

Notice that

${\displaystyle {\frac {10}{2}}={\frac {50}{10}}={\frac {250}{50}}=5}$ i.e., each term divided by the previous one is a non-zero constant.

### Algebraic definition

${\displaystyle \{u_{n}\}}$ is geometric ${\displaystyle \Leftrightarrow {\frac {u_{n+1}}{u_{n}}}=r}$ for all positive integers ${\displaystyle n}$ where${\displaystyle r}$ is a constant (the common ratio)

### The 'Geometric' Mean

If ${\displaystyle a}$,${\displaystyle b}$and${\displaystyle c}$ are any consecutive terms of a geometric sequence then

${\displaystyle {\frac {b}{a}}={\frac {c}{b}}}$ {equating common ratios}

Therefore

${\displaystyle b^{2}=ac}$ and so ${\displaystyle b=\pm {\sqrt {ac}}}$ where ${\displaystyle {\sqrt {ac}}}$ is the geometric mean of ${\displaystyle a}$ and ${\displaystyle c}$ .

### The General Term

Suppose the first term of a geometric sequence is ${\displaystyle u_{1}}$ and the common ratio is${\displaystyle r}$.

Then ${\displaystyle u_{2}=u_{1}r}$therefore${\displaystyle u_{3}=u_{1}r^{2}}$etc.

Thus${\displaystyle u_{n}=u_{1}r^{n-1}}$

 ${\displaystyle u_{1}}$ is the first term of the sequence. ${\displaystyle n}$ is the general term ${\displaystyle r}$ is the common ratio between terms in an geometric sequence.

## Compound Interest

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.,

${\displaystyle 100\%+10\%=110\%}$

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

${\displaystyle \1000\times 1.1=\1100}$

 After two years it is worth After three years it is worth ${\displaystyle \1100\times 1.1}$ ${\displaystyle \1210\times 1.1}$ ${\displaystyle =\1000\times 1.1\times 1.1}$ ${\displaystyle =\1000\times (1.1)^{2}\times 1.1}$ ${\displaystyle =\1000\times (1.1)^{2}}$ ${\displaystyle =\1000\times (1.1)^{3}}$ ${\displaystyle =\1210}$ ${\displaystyle =\1331}$

Note

 ${\displaystyle u_{1}=\1000}$ = The initial investment ${\displaystyle u_{2}=u_{1}\times 1.1}$ = The amount after 2 year ${\displaystyle u_{3}=u_{1}\times (1.1)^{2}}$ = The amount after 3 years ${\displaystyle u_{4}=u_{1}\times (1.1)^{3}}$ = The amount after 4 years ${\displaystyle \vdots }$ ${\displaystyle u_{n+1}=u_{1}\times (1.1)^{n}}$ = amount after n years

In general, ${\displaystyle u_{n+1}=u_{1}\times r^{n}}$is used for compound growth, where

 ${\displaystyle u_{1}}$ is the initial investment ${\displaystyle r}$ is the growth multiplier ${\displaystyle n}$ is the number of years ${\displaystyle u_{n+1}}$ is the amount after ${\displaystyle n}$ years