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Subject: Mathematics

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21 January 2021

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Suitable for GCSE Students sitting the 9 – 1, A-Level Maths students revision and some Key Stage 3 students.

Suitable for all the big examination boards including AQA, Edexcel, OCR and WJEC.

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## Sequences and Nth Term Revision

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In maths, a sequence is a list of numbers, algebraic terms, shapes, or other mathematical objects that follow a pattern or rule. There are two different ways you will be expected to work out a sequence:

• A term-to-term rule – each term in the sequence is calculated by performing a fixed set of operations (such as “multiply by 2 and add 3 ”) to the term(s) before it.
• A position-to-term rule – each term in the sequence is calculated according to its position in the sequence. Usually, this takes the form of an n^{th} term formula.

## Finding a Sequence from the n^{th} Term

Some sequences have a rule called an n^{th} term rule , which tells you how to generate terms.

The n^{th} term rule is always an expression in n .

To find the first term in the sequence , substitute n=1 into the expression . To find the second term in the sequence , substitute n=2 into the expression . And so on.

Example: A sequence has the n^{th} term rule 2n+1 . Find the first four terms .

First term: n=1

Second term: n=2

Third term: n=3

Fourth term: n=4

The first four terms are: 3 , 5 , 7 and 9 .

## Finding the n^{th} term for a Linear Sequence

Linear sequences (or arithmetic progressions) are sequences that increase or decrease by the same amount between each term.

Want a way to express any term in a concise mathematical way? This can be done using the n^{th} term formula. This is a rule that gives you the value of any term in the sequence in the form,

Where a and b are numbers to be determined.

Example: Find the n^{th} term for the following sequence,    3, \, 7, \,11, \,15, \,19, \, ... Step 1: Find the Common Difference ( a )

The  common difference  is the amount the sequence increases (or decreases) each time.

a=4 , because a is always the difference between each term.

Step 2: Determine if you need to Add or Subtract anything ( b )

To work out b , consider the sequence formed by putting n=1, 2, 3, 4, 5 into 4n :

4, 8, 12, 16, 20

What’s the difference between these terms and our actual sequence? They’re all too big by 1 . So, to make our original sequence, we must subtract 1 from 4n .

Step 3: Write the formula in the correct form ( an+b )

Thus, our n^{th} term formula is

## Geometric Sequences

Another type of sequence is a geometric sequence or geometric progression.

In a geometric sequence , you multiply each term by a common ratio to get to the next term. For example, is a geometric progression where to get get to the next term you have to multiply the previous term by the common ratio. Therefore to find the next two terms of this sequence we have to multiply the preceding term by 3 , so

40.5\times 3=121.5 and  121.5\times 3 =364.5

## Geometric Sequences involving Surds is a geometric progression where the common ratio is not a rational number. The method of finding the next two terms of this sequence is the same as before, multiply the preceding term by \sqrt{3} , so

9\sqrt{3}\times 3=27 and  27\times \sqrt{3} = 27\sqrt{3}

## Other Types of Sequences

There are other types of sequences you should be familiar with:

## Triangular Numbers

Triangular numbers are numbers that can be represented as an equilateral triangle of dots. The n^{th} term is \dfrac{n(n+1)}{2} , giving

1, 3, 6, 10, 15, 21, ...

## Square Numbers

These sequences are made up of square numbers so the n^{th} term is n^2 , giving

1, 4, 9, 16, 25, 36, 49, ...

## Cubic Numbers

These sequences are made up of cubic numbers so the n^{th} term is n^3 , giving

1, 8, 27, 64, 125, 216, ...

## The Fibonacci Numbers

The first few terms of the Fibonacci sequence are:

1, 1, 2, 3, 5, 8, 13, 21, ...

This is a famous sequence that you need to recognise. The rule is the previous 2 terms are added together in the sequence to get the next term.

Subscript notation can be used to denote position to term and term to term rules.

For example

x_{n+1}=3x_n-4 shows that you multiply the previous term by 3 and subtract 4 to get the next term.

y_n=5n+3 gives you a rule for the n th term of a sequence.

## Example 1: Linear Sequences

Find the n^{th} term formula for the sequence -2, 5, 12, 19, 26 .

The first step is to find the common difference between each term. Hence we can write the n^{th} term as,

To work out b , consider the sequence formed by putting n=1, 2, 3, 4, 5 into u_n=7n :

7, 14, 21, 28, 35

The difference between these numbers and our sequence is we need to subtract 9 from each term. Thus, our n^{th} term is

## Example 2: Determining if a Value is Part of a Sequence

The first five terms of a sequence are:

-3, 1, 5, 9, 13

Determine if 1143 is part of this sequence. Step 1: First we must find the n^{th} term of the sequence of the sequence as before, this gives

Step 2:  Next we need to write the n^{th} term as an euqation equal to 1143 and solve for n

If n solves to give an integer, then 1143 is part of the sequence.

\begin{aligned} 4n-7&=1143 \\ 4n & = 1150 \\ n&= 287.5\end{aligned}

As 287.5 is not an integer, 1143 must not be part of the sequence.

## Example 3: Solving Problems involving Sequences

The sum of two consecutive terms in a sequence given by the n^{th} term, 3n+8 is 109 . Find the values of these two terms.

In this case we have to first set up an equation, setting the first term as n and the second as n+1 , such that,

\begin{aligned} 3n+8+3(n+1)+8 & = 109 \\ 3n + 8 + 3n + 3 + 8 & = 109 \\ 6n +19 &= 109 \\ 6n &= 90 \\ n &=15 \end{aligned}

Hence, it is the 15 th and 16 th terms we are looking for which are,

3(15)+8 = 53 and 3(16) +8 = 56

## Sequences and Nth Term Example Questions

Question 1: A sequence has the n^{th} term 4n+1

a) Find the 12^{th} term in this sequence

b) A term in this sequence is 77 . Find the position of this term in the sequence.

a) To find the 12^{th}  term of this sequence, we will substitute n=12 into the formula given.

So, the 12^{th} term is 49

b) Every term in this sequence is generated when an integer value of n is substituted into 4n+1

Hence if we set 77 to equal 4n+1 , we can determine its position in the sequence,

making n the subject by subtracting 1 then dividing by 4 ,

n=\dfrac{77-1}{4}=19

Hence 77 is the 19^{th} term in the sequence.  Gold Standard Education

Question 2: A sequence is defined by the formula 5n-4

a) Work out the first 5 terms of this sequence.

b) Explain why 108 is not a term in this sequence.

a) To generate the first 5 terms of this sequence, we will substitute n=1, 2, 3, 4, 5 into the formula given.

\begin{aligned}1 &=5(1)-4 =1 \\ 2 &=5(2)-4 =6 \\ 3 & =5(3)-4=11 \\ 4 &=5(4)-4=16 \\ 5 &=5(5)-4=21 \end{aligned}

So, the first 5 terms are 1 , 6 , 11 , 16 , and 21

b) Every term in this sequence is generated when an integer value of n is substituted into 5n-4 .

If we set 108 to equal 5n-4 , we can determine if it is a part of the sequence or not. If the value of n is a whole number then it is part of the sequence, hence

making n the subject by adding 4 then dividing by 5 ,

n=\dfrac{112}{5}=22.4

As there is no “ 22.4^{th} ” position in the sequence, it must the case that 108 is not a term in this sequence.

Question 3: The first 5 terms of an arithmetic progression are

-3,\,\,2,\,\,7,\,\,12,\,\,17

Find the formula for the n^{th} term of this sequence.

We are told it is an arithmetic progression and so must have n^{th} formula: an+b . To find a , we must inspect the difference between each term which is 5 , hence a=5 .

Then, to find b , let’s consider the sequence generated by 5n :

5,\,\,10,\,\,15,\,\,20,\,\,25

Every term is this sequence is bigger than the corresponding terms in the original sequence by 8 . So, to get to the original sequence, we will have to subtract 8 from every term in this sequence. In other words, the n^{th} term formula for our sequence in question is

## Sequences and Nth Term Worksheet and Example Questions ## (NEW) Sequences and Nth Term (Linear) Exam Style Questions - MME

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This topic is relevant for: Here we will learn about different types of sequences including arithmetic sequences, geometric sequences and quadratic sequences and how to generate them and find missing terms, along with special sequences like the fibonacci sequence. We will also learn how to find the nth term of linear sequence and the nth term of a geometric sequence and how to work out whether a particular number appears in a sequence.

There are also sequences worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is a sequence?

A number sequence is a set of numbers that follow a particular pattern or rule to get  from term to term.

There are four main types of different sequences you need to know, they are arithmetic sequences , geometric sequences , quadratic sequences and special sequences. 1 Arithmetic sequences

An arithmetic sequence is an ordered set of numbers that have a common difference between each term.

If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence .

Step-by-step guide: Arithmetic Sequence

## Arithmetic sequences examples

Example 1: sequence with a term to term rule of +3..

We add three to the first term to give the next term in the sequence, and then repeat this to generate the sequence.

## Example 2: sequence with a term to term rule of -1.

We subtract 1 from the first term to give the next term in the sequence, and then repeat this to generate the sequence.

We can work out previous terms by doing the opposite of the term to term rule.

2 Geometric sequences

A geometric sequence is an ordered set of numbers that progresses by multiplying or dividing each term by a common ratio.

If we multiply or divide by the same number each time to make the sequence, it is a geometric sequence .

Step-by-step guide: Geometric sequences

## Geometric sequences examples

Example 3: sequence with a term to term rule of ×2..

We multiply the first term by 2 to give the next term in the sequence, and then repeat this to generate the sequence.

## Example 4: sequence with a term to term rule of ÷2.

We divide the first term by 2 to give the next term in the sequence, and then repeat this to generate the sequence.

A quadratic sequence is an ordered set of numbers that follow a rule based on the sequence n 2 = 1, 4, 9, 16, 25, … (the square numbers).

The difference between each term is not equal, but the second difference is.

Example 5: sequence with common second difference of +2..

We find the first difference of the sequence and then find the term to term rule for the second difference. The second difference will always be the same for quadratic sequences.

4 Special sequences

You will need to be able to recognise some important special sequences. ## Special sequences examples

Example 6: square numbers.

A square number is the result when a number is multiplied by itself.

E.g. 1×1=1, 2×2=4, 3×3=9 etc .

The square numbers can form a sequence: 1, 4, 9, 16, 25, 36, 49…

n th Term = n 2

## Example 7: cube numbers

A cube number is the result when a number is multiplied by itself three times.

1×1×1=1, 2×2×2=8, 3×3×3=27 etc.

The cube numbers can form a sequence: 1, 8, 27, 64, 125 …

n th Term = n 3

## Example 8: triangular numbers

The triangular numbers as numbers that can form a triangular dot pattern. They are also special type of quadratic sequence.

We can generate a sequence of triangular numbers by adding one more to the term to term rule each time:

## Example 9: Fibonacci numbers

We can generate the Fibonacci Sequence of numbers by adding the previous two numbers together to work out the next term.

First and second terms :

0+1=1 , so the third term is 1 .

Sequence : 0, 1, 1

Fourth Term :

Sequence : 0, 1, 1, 2

Fifth Term :

Sequence : 0, 1, 1, 2, 3

We can continue to follow the pattern to generate an infinite sequence.

The Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21 , …

The Fibonacci Sequence forms a spiral that is seen throughout nature.

## What are the four main types of different sequences? ## Sequence rule to find a term

We use the n th term of a sequence to work out a particular term in a sequence. By substituting in the number of the term we want to find as ‘ n ’ we can generate the specific term in the sequence.

What is the n th term and the 21 st term of this sequence?

4, 10, 16, 22, …

The n th term of this sequence is 6n – 2 .

To find the 21 st term, n = 21

(6 x 21) – 2 = 124

The 21 st term is 124.

Given the n th term rule, 3n + 4 , find the 20 th and 100 th term for this sequence.

To find the 20 th term, n = 20

(3 x 20) + 4 = 64

To find the 100 th term, n = 100

(3 x 100) + 4 = 304. The 20 th term is 64 and the 100 th term is 304.

## How to find the nth term of a sequence

The n th term is a formula that enables us to find any term in a sequence.

We will need to be able to find the n th term of a linear (arithmetic) sequence , and the n th term of a quadratic sequence .

We can make a sequence using the nth term by substituting different values for the term number n into it.

Step-by-step guide: Nth term of a sequence ## Sequences worksheet

Get your free sequences worksheet of 20+ questions and answers. Includes reasoning and applied questions.

1 nth term of a linear sequence

In order to find the n th term of a linear sequence:

• Step 1: find the common difference for the sequence.
• Step 2: multiply the values for n = 1, 2, 3 , … by the common difference.
• Step 3: add or subtract a number to obtain the sequence given in the question.

## How to find the nth term of a linear sequence ## nth term of a linear sequence example

Find the n th term for the sequence 5, 9, 13, 17, 21 , ….

Find the common difference for the sequence.

The common difference d = 4 .

Multiply the values for n = 1, 2, 3 , … by the common difference.

Here, we generate the sequence 4 n = 4, 8, 12, 16, 20 , …. (the 4 times table).

Add or subtract a number to obtain the sequence given in the question.

The n th term of this sequence is 4n+1 .

2 nth term of a quadratic sequence

In order to find the n th term of a quadratic sequence we have to find the second difference. To do this, we calculate the first difference between each term and then calculate the difference between this new sequence.

• Step 1: find the first difference (d 1 ) and second difference (d 2 ) for the sequence.
• Step 2: Halve the second difference to find a , the coefficient of n 2 .
• Step 3: Subtract an 2 from the original sequence.
• Step 4: If this produces a linear sequence, find the n th term of it.
• Step 5: Add the n th term for the linear sequence to an 2 to work out the n th term of the quadratic sequence.

## How to find the nth term of a quadratic sequence ## nth term of a quadratic sequence example

Find the n th term rule of the quadratic sequence:

5, 14, 29, 50, 77 , …

Find the first difference (d 1 ) and second difference (d 2 ) for the sequence.

The second difference d 2 =6.

Halve the second difference to find a the coefficient of n 2 .

As d 2 = 6 ,

d 2 ÷ 2 = 6 ÷ 2 = 3

This means a = 3 and so we have the sequence 3n 2 .

Subtract 3n 2 from the original sequence. Here, the remainder for each term is 2 .

Find the nth term for the linear sequence generated.

Not required for this example as the remainder is 2 for each term.

Add the nth term for the linear sequence to 3n 2 to find the nth term of the quadratic sequence.

The nth term of the quadratic sequence is 3 n 2 +2.

3 Use the nth term to calculate any term in a sequence

We can calculate any term in a sequence by substituting the term number into the n th term.

## nth term to calculate any term in a sequence example

Example 10: find terms in the sequence with nth term = 4n+1.

• For the first term, n=1

So 1st term = 4 ✕ 1 + 1 = 5

• For the 10th term, n=10

So 10th term = 4 ✕ 10 + 1= 41

• For the 100th term, n=100

So 100th term = 4 ✕ 100 + 1 = 401

## Example 11: find the first three terms in the sequence n 2 + 7n

• For the first three terms we use n=1, n=2 and n=3:

4 Use the nth term to work out whether a number is in a sequence

We can use the n th term to work out whether a number is in a sequence by putting the n th term equal to the number and solving the equation to find n .

Because n is the term number it has to be an integer (a whole number).

i.e. for the 1st term n=1 , for the 9th term n=9 etc .

When n is an integer then the number is in the sequence.

## nth term to work out whether a number is in a sequence examples

Example 12: when n is an integer then the number is in the sequence.

Is 25 a number in the sequence generated by the n th term 4n+1 ?

Let’s put 25 equal to 4n+1 and solve the equation.

As 6 is an integer this means that 25 is the 6th term in the sequence.

So 25 is a number in the sequence.

## Example 13: when n is not an integer then the number is not in the sequence

Is 60  a number in the sequence generated by the n th term 4n+1 ?

Let’s put 60 equal to 4n+1 and solve the equation.

As 14.75 is not an integer this means that 60 is not a number in the sequence.

5 Solve problems using algebra in sequences

We can write any type of sequence described above using algebraic terms rather than numerical terms.

For example, the sequence,

is an arithmetic sequence, because we add 2 each time to get from one term to the next.

We can apply skills such as solving linear equations and solving simultaneous equations to sequences with algebraic terms.

## Example 14: a Fibonacci sequence

The first three terms of a Fibonacci sequence are,

The 3rd term is 12.

Find the value of the 4th term.

First we form and solve an equation using the information given about the 3rd term:

We also need to find an algebraic expression for the 4th term. To get the next term in a Fibonacci sequence, we add the two previous terms. So the 4th term will be the sum of the 2nd and 3rd terms:

To find the value of the 4th term, substitute a=5 to get 3(5)+4=19 .

So the 4th term is 19.

Step-by-step guide: Sequences algebra

## Common misconceptions

• Arithmetic sequences are generated by adding or subtracting the same amount each time – they have a common difference d .
• Geometric sequences are generated by multiplying or dividing by the same amount each time – they have a common ratio r .
• Quadratic sequences have a common second difference d 2 .
• Mixing up working out a term in a sequence with whether a number appears in a sequence

In order to find any term in a sequence using the n th term we substitute a value for the term number into it.

• For the 5th term n=5 ,

So 5th term= 4(5)+1=21

• For the 8th term n=8

So 8th term= 4(8)+1=33

In order to work out whether a number appears in a sequence using the n th term we put the number equal to the n th term and solve it. If n (the term number) is an integer the number is in the sequence, if n is not an integer the number is not in the sequence.

Is 22 a number in the sequence with nth term = 4n+1 ?

As 5.25 is not an integer this means that 22 is not a number in the sequence.

## Practice sequences questions

1. Find the nth term of 1, 5, 9 ,13, … The common difference here is 4 so it is 4n . To get from 4n to our sequence we need to subtract 3 so our sequence is 4n-3 .

2. Find the nth term of 7, 4, 1,-2,-5, …

The common difference here is -3 so it is -3n . To get from -3n to our sequence we need to add 10 so our sequence is -3n+10 .

3. Find the nth term of 6, 17, 32, 51, 74

Here the second difference is 4 . 4 \div 2 = 2

So it is 2n^{2} We now need to work out the nth term formula for 4, 9, 14, 19, 24 ,..

The difference is 5 so it is 5n . To get from 5n to our sequence we need to subtract 1 so it is 5n-1 .

The overall formula for the sequence is 2n^{2}+5n-1.

4. Generate the first 6 terms of the sequence 3n-4 5. Work out the 10 th term of the sequence 25-n^{2}

n=10   \begin{aligned} 25-10^{2}&=25-100\\ &=-75\end{aligned}

6. Write down the next three terms in this sequence: 0, 1, 1, 2, 3, 5, 8, …

This is the Fibonacci Sequence. We add the two previous terms to get the next term:

## \begin{aligned} 5+8=13\\ 8+13=2\\ 13+21=34 \end{aligned}

Sequences gcse questions.

1. (a) 5, 8, 11, 14

What is the rule to get from one term to the next?

(b) What is the n th term of the sequence?

2.(a)What is the 12 th term in the sequence 2n-20 ?

(b) Is 180 a number in the sequence?

## \begin{aligned} 2n-20&=180\\ 2n&=200\\ n&=100 \end{aligned}

Yes because n is an integer. 180 is the 100 th term in the sequence.

3. Look at this sequence 1, 2, 4, 8, 16

(a) What is the common ratio in the above geometric sequence?

(b) What would be the 7 th and 8 th term for the sequence?

7 th term= 32 \times 2 = 64

8 th term= 64 \times 2 = 128

## Learning checklist

You have now learned how to:

• Recognise arithmetic sequences and find the nth term.
• Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci Sequences, quadratic sequences, and simple geometric progressions.
• Generate terms of a sequence from either a term-to-term or a position-to-term rule.
• Recognise and use quadratic sequences.
• Deduce expressions to calculate the nth term of quadratic sequences.

## The next lessons are

• Substitution
• Solving equations

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