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

$3,7,11,15,\ldots$

The above is a type of number sequence. The first term is $3$ , the second is $7$ , 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:

$2,5,8,11,14,\ldots$ is arithmetic as $5-2=8-5=11-8=14-11$ etc

Algebraic Definition

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

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

Where $d$ is defined as:

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

Here, the notation is as follows:

$a_{1}$ is the first term of the sequence.

$n$ is the number of terms in the sequence.

$d$ is the common difference between terms in an arithmetic sequence.

Example

Given the sequence $1,3,5,7,\ldots ,n$ , the values of the notation are as follows:

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

And

$a_{1}=1$

Therefore

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

Thus we can determine any value within a sequence:

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

Arithmetic Series

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

$21+23+27+\cdots +49$

Sum of an Arithmetic Series

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

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

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

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

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

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

Geometric Sequences

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

$2,10,50,250,\ldots$ is geometric as $2\times 5=10$ and $10\times 5=50$ and $50\times 5=250$ .

Notice that

${\frac {10}{2}}={\frac {50}{10}}={\frac {250}{50}}=5$

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

Algebraic definition

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

The 'Geometric' Mean

If $a,b,c$ are any consecutive terms of a geometric sequence, then

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

Therefore

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

The General Term

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

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

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

$a_{1}$ is the first term of the sequence.

$n$ is the general term.

$r$ 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.,

$100\%+10\%=110\%$

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

$1000\$\times 1.1=1100\$$

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

Note

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

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

$a_{1}$ is the initial investment

$r$ is the growth multiplier

$n$ is the number of years

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