Find a Formula for the Nth Term an of the Sequence Assuming That the Indicated Pattern Continues

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Nth Term Of A Sequence

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.

What is the nth term?

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 substitute n = 1 into the nth term.

1st term =  2(1) + 1 = 3

• To find the second term we substitute n = 2 into the nth term.

2nd term =  2(2) + 1 = 5

• To find the third term we substitute n = 3 into the nth term.

3rd term =  2(3) + 1 = 7

• To find the tenth term we substitute n = 10 into the nth term.

10th term =  2(10) + 1 = 21

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

Type of Sequence	Example	nth Term
Arithmetic	6, 2, -2, -6, -10, ...	10-4n
Geometric	1, 2, 4, 8, 16, 32, ...	2{n-1}
Quadratic	3, 9, 19, 33, 51, ...	2n^{2}+1
Cubic	2, 22, 78, 188, 370, ...	3n^{3}-n

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

What is the nth term?

nth term worksheet

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

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nth term worksheet

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

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How to find the nth term

The nth term of an arithmetic sequence is given by :

a_{n}=a_{1}+(n-1) d

To find the nth term, first calculate the common difference, d .

Next multiply each term number of the sequence (n = 1, 2, 3, …) by the common difference.

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 an + b where a and b are unknown values that we will have calculated.

To summarise, in order to find the n^th term of an arithmetic sequence:

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

Explain how to find the nth term of an arithmetic sequence in 3 steps

nth term formula

The nth term formula for an arithmetic sequence is a_n=a_1+(n-1)d

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:

a_n=a_1(r)^{n-1}

Where,

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

a_{1} is the first term

r is the common ratio

• The nth term formula for a quadratic sequence is:

an^2+bn+c

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

nth term examples

Example 1: find the nth term for an increasing arithmetic sequence

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

1. Find the common difference for the sequence.

Here, 9 − 5 = 4.

The common difference d = 4.

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

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

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

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

Example 2: find the nth term for a decreasing sequence, including negative numbers

Find the n^th term for the sequence 3, 1, -1, -3, -5, ….

Find the common difference for the sequence.

Here, 1 − 3 = -2

The common difference d = -2.

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

Here, we generate the sequence -2n = -2, -4, -6, -8, -10, … (the multiples of -2).

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

The n^th term of this sequence is -2n + 5  (or 5 − 2n).

Example 3: find the nth term of a sequence including decimals.

Find the n^th term for the sequence 0.2, 0.5, 0.8, 1.1, 1.4, ….

Find the common difference for the sequence.

Here, 1.1 − 0.8 = 0.3

The common difference d = 0.3.

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

Here, we generate the sequence 0.3n = 0.3, 0.6, 0.9, 1.2, 1.5, … (the multiples of 0.3).

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

The

n^th

term of this sequence is 0.3n − 0.1 or

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

Example 4: find the nth term of a sequence including negatives and decimals

Find the n^th term for the sequence -9.1, -8.3, -7.5, -6.7, -5.9, ….

Find the common difference for the sequence.

Here, -8.3 − (-9.1) = -8.3 + 9.1 = 0.8

The common difference d = 0.8.

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

Here, we generate the sequence 0.8n = 0.8, 1.6, 2.4, 3.2, 4, … (the multiples of 0.8).

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

The n^th term of this sequence is 0.8n − 9.9 .

Example 5: find the nth term of a sequence including fractions

Find the n^th term for the sequence

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

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 n^th term of this sequence is

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

Example 6: find the nth term when interpreting a pattern

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

Find the common difference for the sequence.

By counting the number of sides we can see that the first term in the sequence is 12.

The second term in the sequence is 21.

The next term 30.

Here, 21 − 12 = -9

The common difference d = 9.

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

Here, we generate the sequence 9n = 9, 18, 27, 36, 45, … (the multiples of 9).

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

The n^th term of this sequence is 9n + 3.

Common misconceptions

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

For example, the sequence 3, 6, 9, 12, 15, … has the n^th term 3n but is incorrectly written as n + 3.

• The n ^th term is incorrectly simplified

For example, if the n^th term of a sequence is equal to 6n − 4, the solution would be incorrectly simplified to 2n.

• For decreasing sequences, the n ^th term has a positive common difference

For example, taking the decreasing sequence -2, -4, -6, -8, -10, … which has the n^th term of -2n but is incorrectly stated as 2n which would be an increasing sequence. This is also true with the constant.

Practice nth term questions

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

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 .

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 .

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 .

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 .

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 .

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

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

nth term GCSE questions

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)

Learning checklist

You have now learned how to:

• Recognise arithmetic sequences
• Find the n^th term

Still stuck?

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