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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.

For quadratic sequences, please visit the Quadratic Sequences Revision Page

## 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

You may be asked about geometric sequences involving surds. For example,

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.

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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

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## GCSE MATHS sequences worksheet

Subject: Mathematics

Age range: 14-16

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Last updated

21 January 2021

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A GCSE Maths Worksheet covering sequences and the nth term

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## Sequences at a glance

Sequences are a set of numbers that have a starting number and go on forever. Number sequences can increase or decrease and can be whole numbers, fractions or decimals.

There are different ways to classify a sequence including arithmetic sequences and geometric sequences. Arithmetic sequences have a common difference and geometric sequences have a common ratio. The common difference or common ratio can be used to continue a sequence, or to find missing terms within the sequence and may be referred to as the term-to-term rule.

Arithmetic sequences are based on multiples which can be used to find the nth term rule (or nth term formula) by calculating the common difference by subtracting two consecutive terms in the sequence. The common difference gives the coefficient of the n in the nth term formula.

The nth term can be referred to as the position-to-term rule and can be used to find a particular term in the sequence or used to check if a particular value is in the sequence or not. A sequence that is increasing or decreasing by the same amount is called a linear sequence.

There are other types of sequences such as the square numbers that form a sequence which is involved in quadratic sequences. Other non-linear sequences include the cube numbers and the powers of 10. There is also the Fibonacci sequence where the next term is found by adding the previous two terms. Triangular numbers is another special sequence which is useful to know.

Looking forward, students can progress with other sequences worksheets and on to additional algebra worksheets , for example a simplifying expressions worksheet , straight line graph or simultaneous equations worksheet .

For more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts.

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