# Sequences

## Generating sequences

Sequences are numbers that follow a particular pattern. This means the numbers are connected in some way.

To generate sequences we use an equation or a rule.

The **term to term** rule.

When you generate a sequence from a term to term rule, you use the one term to generate another. Apply the rule to one term to generate the following term in the sequence.

For example starting on 5, generate a sequence using the term to term rule of +9. This means to go from one term in my sequence to another I need to add 9

The first term in our sequence has the term number 1. The term number helps us refer to the place number is in the sequence.

**Position to term rule**

If we generate sequences using a position to term rule you can generate any term in the sequence. This is often referred to as the nth term.

For example, if a sequence has the position to term rule (or nth term) of 4n +2 , the 10th term in the sequence will be when n equals 10:

=4n +2

=4(10) +2

=40+2

=42 therefore the 10th term in this sequence is 42.

## Recognising different sequences

There are a couple of sequences that are useful to know:

**Linear sequences:** these sequences increase or decrease by the same amount each time. Eg. 10,13,16, each time the numbers in this sequence increase by +3

These sequences are also known as arithmetic sequences.

**Triangular numbers:** 1, 3 ,6, 10, 15, 21….. triangular numbers, continue adding the natural numbers (1,2,3,4,5….)

**Square numbers:** are the term number square: 1, 4 ,9, 16, 25…

__Cube numbers: __are the term number cubed: 1, 8, 27, 64

**Fibonacci sequence:** 1, 1, 2, 3, 5, 8… this sequence is often found in nature (like these plants).. You add the previous two terms together to find the next term (eg. 1, 1 = 1+1=2- hence 1,1,2…)

**Quadratic sequences**: these sequences will have an nth term that has n^{2} as a part of it. They can be identified by a constant in their second difference.

Eg: 2, 5, 10, 17

The first difference between these numbers is: 3,5,7

The second difference is constant! It’s +2 therefore we have a quadratic!

**Geometric sequences:** these have a common ratio between each term, usually it means that you will have to multiply each term by something to find the next term.

Eg. 10, 100 ,1000, 10000 is a geometric sequence as you need to multiply the previous term by 10 to go from one term to another. Dividing by a constant also shows a geometric sequence: 500, 100, 20,4 - in this sequence we are dividing by 5 to go from one term to the next.

## Finding the nth term

Often you will be required to find the nth term, or the position to term rule of a sequence.

Follow these steps to find the nth term of a **linear** sequence:

Step 1: Look for the term to term rule of the sequence

5, 8, 11, 14….

To go from one term to another you add 3. This means that ( like the 3 times table) 3n is part of the nth term.

Step 2: If we compare the sequence to the times table it is related to, we realise that our first term would have to start on a different number. Therefore we go to the ‘zeroth’ term (i.e. when n is zero), to find the amount we need to add on or take away.

**2**, 5, 8, 11, 14

Because it is a positive 2 we add this to our 3n.

Therefore our nth term is 3n+2

Step 3: Check this by substituting another term number that you already have. For example, we know the 2nd term is 8 - therefore 3(2)+2=8 it does! So we are correct.

This process works exactly the same for negative sequences:

Eg. 10, 5, 0, -5….

Step 1: The common difference is -5 therefore -5n is part of the sequence

Step 2: 10+5 = +15, therefore we add this to -5n making our nth term -5n+15 or 15-5n

Step 3: Check -5(2)+15= 5 therefore we are correct!

## Finding the nth term for a quadratic sequence

When it comes to finding the nth term of a quadratic sequences we have a couple more steps to complete.

Firstly we need to remember the structure of a quadratic sequence: ax^{2} +bx +c

a,b,c are all related by these formula:

The second difference = 2a

The first term = a + b+ c

2nd term - 1st term = 3a+b

Step 1: is the same- find the difference between the numbers in the sequence. We’ll call this the *first difference*

This will not be constant as it is not a linear sequence!

Eg. 5 11 21 35

*First difference* +6, +10, +12

Step 2: Find the difference between the_ first difference_

Eg. 5 11 21 35

_First difference _+6, +10, +14

*Second difference +4 +4*

Step 3: Refer back to these formula

The second difference = 2a the second difference = 4 therefore 2a=4, so **a=2**

2nd term - 1st term = 3a+b

11 -5 =3(2) +b

6 = 6+b

__0=b __

The first term = a + b+ c

5 =2 +0+c

**3=c**

**Therefore our nth term = 2n**^{2}__ + 3__

- What type of sequence is this: 6, 10, 14, 18
- Your answer should include: Linear / Arithmetic