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Substitution Arithmetic sequencesThis topic is relevant for:

Here we will learn about how to find the **n ^{th} term of an arithmetic sequence**. You’ll learn what the nth term is and how to work it out from number sequences and patterns.

At the end you’ll find nth term worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

The nth term is a formula that enables us to find any term in a sequence. The ‘n’ stands for the term number.

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

To find the 10th term we would follow the formula for the sequence but substitute 10 instead of ‘n’; to find the 50th term we would substitute 50 instead of n.

**For example if the nth term = 2n + 1 **

- To find the
**first term**we substituten = 1 into the nth term.

**1st term** **3**

- To find the
**second term**we substituten = 2 into the nth term.

**2nd term** **5**

- To find the
**third term**we substituten = 3 into the nth term.

**3rd term** **7**

To find the tenth term we substitute

**10th term**= 2(10) + 1 = **21**

Below are a few examples of different types of sequences and their nth term formula.

**Type of Sequence**

Arithmetic

Geometric

Quadratic

Cubic

**Example**

6, 2, -2, -6, -10, …

1, 2, 4, 8, 16, 32, …

3, 9, 19, 33, 51, …

2, 22, 78, 188, 370, …

**N ^{th} Term**

10 − 4n

2^{n -1}

2n^{2} + 1

3n^{3} − n

In this lesson, we will look specifically at finding the n^{th} term for an arithmetic or linear sequence.

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

DOWNLOAD FREEGet your free nth term worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREETo find out more about the different types of sequences, and how to answer sequence related questions you may find it helpful to look at the introduction to sequences lesson or one of the others in this section.

- Sequences
- Arithmetic sequence
- Quadratic sequences
- Quadratic nth term
- Geometric sequence

The nth term of an arithmetic sequence is given by :

a_{n}=a_{1}+(n-1) dTo find the nth term, first calculate the **common difference**, **d**

Next multiply each term number of the sequence

Then add or subtract a number from the new sequence to achieve a copy of the sequence given in the question.

This will give you the nth term term in the form

To summarise, in order to find the ^{th}

- Find the common difference for the sequence.
- Multiply the values for
n = 1, 2, 3 . - Add or subtract a number to obtain the sequence given in the question.

- The
**nth term formula**for an**arithmetic sequence**is:

Where,

a_{n} is the n^{th} term (general term)

a_{n} is the first term

n is the term position

d is the common difference

- The
**nth term formula**for a**geometric sequence**is:

Where,

a_{n} is the n^{th} term (general term)

a_{1} is the first term

n is the term position

r is the common ratio

- The
**nth****term formula**for a**quadratic sequence**is:

Where,

a, b and c are constants (numbers on their own)

n is the term position

a + b + c is the first term

3a + b is the first difference between

2a is the second difference

Find the ^{th}

- Find the common difference for the sequence.

Here,

The common difference

2 Multiply the values for

Here, we generate the sequence

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

The ^{th}** 4 n + 1**.

Find the ^{th}

Find the common difference for the sequence.

Here,

The common difference

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

Here, we generate the sequence

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

The ^{th}**-2n + 5** (or **5 − 2n**)

Find the ^{th}

Find the common difference for the sequence.

Here,

The common difference

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

Here, we generate the sequence

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

The ^{th}** 0.3n − 0.1 **or

\[\frac{3 n-1}{10}\]

Find the ^{th}

Find the common difference for the sequence.

Here,

The common difference

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

Here, we generate the sequence

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

The ^{th}**0.8n − 9.9**

Find the ^{th}

\[\frac{1}{4}, \frac{5}{8}, 1,1 \frac{3}{8}, 1 \frac{3}{4}, \ldots\]

Find the common difference for the sequence.

Here

\[\frac{5}{8}-\frac{1}{4}=\frac{5}{8}-\frac{2}{8}=\frac{3}{8}\]

The common difference

\[d=\frac{3}{8}\]

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

Here, we generate the sequence

\[\frac{3 n}{8}=\frac{3}{8}, \frac{3}{4}, 1 \frac{1}{8}, 1 \frac{1}{2}, 1 \frac{7}{7}, \ldots\]

\[\left ( \text{the multiples of }\frac{3}{8} \right ).\]

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

The ^{th}

\[\frac{3 n}{8}-\frac{1}{8} \text { or } \frac{3 n-1}{8}\]

Using the patterns below, write an expression for the number of lines in pattern

Find the common difference for the sequence.

By counting the number of sides we can see that the first term in the sequence is

The second term in the sequence is

The next term

Here,

The common difference

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

Here, we generate the sequence

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

The ^{th}

**The common difference is used as the constant instead of the multiplier**

For example, the sequence ^{th}

**The**n ^{th}term is incorrectly simplified

For example, if the ^{th}

**For decreasing sequences, the**term has a positive common differencen ^{th}

For example, taking the decreasing sequence ^{th}

1. Write down the first three terms in the sequence 4n-7 .

4, -3, -10

-3, -10, -17

4, 11, 18

-3, 1, 5

\begin{array}{l} 4 \times 1 – 7 = -3\\\\ 4 \times 2 – 7 = 1 \\\\ 4 \times 3 -7 = 5 \end{array}

2. Below is a table describing the position of each term in an arithmetic sequence and the value of these terms.

State the value of the first term in the sequence.

\begin{aligned} &\quad n \quad \quad 1 \quad \quad2 \quad \quad 3 \quad \quad 4 \quad \quad 5 \\ &an + b \quad \quad \quad 8.7 \quad 15.9 \quad \quad \quad 30.3 \end{aligned}

7.2

0

23.1

1.5

It is an arithmetic sequence meaning the difference between each term is the same.

15.9-8.7 = 7.2 so the difference between each term is 7.2 .

8.7-7.2=1.5 therefore the first term is 1.5 .

3. Below are the first 5 terms of an arithmetic sequence.

8, \quad 13, \quad 18, \quad 23, \quad 28, …Find the n^{th} term of the sequence.

n+5

5n-3

5n+3

2n

The common difference here is 5 so it is 5n .

To get from 5n to our sequence we need to add 3 so our sequence is 5n+3 .

4. Find the n^{th} term formula of the sequence:

-10, \quad-20, \quad -30 \quad -40 \quad -50, …

10n-20

-10n-10

-10n

-10n+10

The common difference is -10 so it is -10n .

We do not need to add or subtract anything here so the nth term is just -10n .

5. The number of petals on a sunflower can be represented as a linear sequence.

Write an expression for the number of petals on sunflower n .

2n+5

2n+3

2n-3

5n

The number of petals on the first three flowers are 5, 7 and 9 .

We need to find the n^{th} term of this sequence.

The common difference is 2 so it is 2n .

We need to add 3 to the sequence 2n so the expression is 2n+3 .

6. The number of square tiles around a pool generates an arithmetic sequence.

How many tiles would there be around a pool of width 30 ?

124

120

240

116

Around the first three pools, the number of tiles are 8, 12 and 16 .

The n^{th} formula for this sequence is 4n+4.

Substituting n = 30 , 4 \times 30 + 4 = 124 .

7. Find the n^{th} term of the linear sequence:

2 \frac{1}{3}, \; \; 2 \frac{2}{3}, \; \; 3, \; \; 3 \frac{1}{3}, \; \; 3 \frac{2}{3}, \ldots

\frac{n}{3}+2

2n+ \frac{1}{3}

3n-2

2n+3

The common difference is \frac{1}{3} so it is \frac{1}{3} n .

Another way of writing this is \frac{n}{3} .

1. A sequence of patterns is made using triangles.

(a) What is the n^{th} term formula for the number of triangles?

(b) How many dark purple triangles would there be in pattern number 100 ?

**(3 marks)**

Show answer

(a)

Sequence 1, 3, 5, 7 – common difference is 2

**(1)**

2n – 1

**(1)**

(b)

99 (one less than the pattern number)

**(1)**

2. (a) Write down an expression for the n^{th} term of the following sequence:

-4, -1, 2, 5, 8, ….

(b) Is the number 101 in this sequence? Show how you decide.

**(4 marks)**

Show answer

(a)

Common difference is 3

**(1)**

3n-7

**(1)**

(b)

3n − 7 = 101

**(1)**

\begin{aligned} 3n&=108\\\\ n&=36 \end{aligned}

Yes it is the 36th term

**(1)**

3. The n^{th} of a sequence is 2n + 3 .

The n^{th} of a different sequence is 5n − 2 .

There are two numbers under 30 that appear in both sequences. What are the two numbers?

**(3 marks)**

Show answer

2n + 3: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29,..

**(1)**

5n − 2: 3, 8, 13, 18, 23, 28, …

**(1)**

13 and 23

**(1)**

You have now learned how to:

- Recognise arithmetic sequences
- Find the n
^{th}term

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