# 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

$3, 7, 11, 15, ...$

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:

$2, 5, 8, 11, 14, ...$ 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:

$u_n = u_1 + (n-1)d$

Where $d$ is defined as:

$d = u_{n+1} - u_n$

Here, the notation is as follows:

 $u_1$ 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 $1, 3, 5, 7 ... n$, the values of the notation are as follows:

$d = u_{n+1} - u_n$

$d = u_2 - u_1 = 3 - 1$

$d = 2$

And

$u_1 = 1$

Therefore

$u_n = u_1 + (n-1)d$

$u_n = 1 + (n-1)2$

$u_n = 1 + 2n - 2$

$u_n = 2n - 1$

Thus we can determine any value within a sequence:

$u_5 = 2(5) - 1 = 10 - 1 = 9$

## Arithmetic Series

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

$21 + 23 + 27 + ... + 49$

### Sum of an Arithmetic Series

Recall that if the first term is $u_1$ and the common difference is $d$, then the terms are:

$u_1, u_1 + d, u_1 + 2d, u_1 + 3d, ...$

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

$S_n = u_1 + (u_1 + d) + (u_1 + 2d) + ... + (u_n - 2d) + (u_n - d) + u_n$

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

$S_n = u_n + (u_n - d) + (u_n - 2d) + ... + (u_1 + 2d) + (u_1 + d) + u_1$

Then add the two sequences together:

$2S_n = (u_1+u_n) + (u_1 + u_n) + (u_1 + u_n) + ... + (u_1 + u_n) + (u_1 + u_n) + (u_1 + u_n)$

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

$2S_n = n(u_1 + u_n)$

$S_n = \frac{n}{2}(u_1 + u_n)$

## Geometric Sequences

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

$2, 10, 50, 250, ...$ 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-zero constant.

### Algebraic definition

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

### The 'Geometric' Mean

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

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

Therefore

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

### The General Term

Suppose the first term of a geometric sequence is $u_1$ and the common ratio is$r$.

Then $u_2 = u_1r$therefore$u_3=u_1r^2$etc.

Thus$u_n=u_1r^{n-1}$

 $u_1$ 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

 $u_1 = 1000$ = The initial investment $u_2 = u_1\times1.1$ = The amount after 2 year $u_3 = u_1\times(1.1)^2$ = The amount after 3 years $u_4 = u_1\times(1.1)^3$ = The amount after 4 years $\vdots$ $u_{n+1} = u_1 \times (1.1)^n$ = amount after n years

In general, $u_{n+1}=u_1 \times r^n$is used for compound growth, where

 $u_1$ is the initial investment $r$ is the growth multiplier $n$ is the number of years $u_{n+1}$ is the amount after $n$ years