# Sequences and Series

## Number Patterns

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

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

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:

${\displaystyle 2,5,8,11,14,\ldots }$ 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:

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

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

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

Here, the notation is as follows:

${\displaystyle a_{1}}$ 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 ${\displaystyle 1,3,5,7,\ldots ,n}$ , the values of the notation are as follows:

{\displaystyle {\begin{aligned}d&=a_{n+1}-a_{n}\\d&=a_{2}-a_{1}=3-1\\d&=2\end{aligned}}}

And

${\displaystyle a_{1}=1}$

Therefore

{\displaystyle {\begin{aligned}a_{n}&=a_{1}+(n-1)d\\a_{n}&=1+(n-1)2\\a_{n}&=1+2n-2\\a_{n}&=2n-1\end{aligned}}}

Thus we can determine any value within a sequence:

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

## Arithmetic Series

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

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

### Sum of an Arithmetic Series

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

${\displaystyle a_{1},a_{1}+d,a_{1}+2d,a_{1}+3d,\ldots }$

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

${\displaystyle {\begin{matrix}&S_{n}&=&a_{1}&+&(a_{1}+d)&+&(a_{1}+2d)&+&\cdots &+&(a_{n}-2d)&+&(a_{n}-d)&+&a_{n}\\+\\&S_{n}&=&a_{n}&+&(a_{n}-d)&+&(a_{n}-2d)&+&\cdots &+&(a_{1}+2d)&+&(a_{1}+d)&+&a_{1}\\\hline \\&2S_{n}&=&(a_{1}+a_{n})&+&(a_{1}+a_{n})&+&(a_{1}+a_{n})&+&\cdots &+&(a_{1}+a_{n})&+&(a_{1}+a_{n})&+&(a_{1}+a_{n})\end{matrix}}}$

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

${\displaystyle {\begin{matrix}2S_{n}=n(a_{1}+a_{n})\\S_{n}={\dfrac {n(a_{1}+a_{n})}{2}}\end{matrix}}}$

## Geometric Sequences

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

${\displaystyle 2,10,50,250,\ldots }$ 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-0 constant.

### Algebraic definition

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

### The 'Geometric' Mean

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

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

Therefore

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

### The General Term

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

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

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

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

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

${\displaystyle a_{1}}$ is the initial investment

${\displaystyle r}$ is the growth multiplier

${\displaystyle n}$ is the number of years

${\displaystyle a_{n+1}}$ is the amount after ${\displaystyle n}$ years