The n^th term rule can calculate the position of the value that is equal to the given value.
If this is not an integer, then you can round down or up as needed.
This term number can be used to generate the desired value.

Keywords

Arithmetic sequence - An arithmetic (or linear) sequence is a sequence where the difference between successive terms is a constant.
N^th term - The n^th term of a sequence is the position of a term in a sequence where n stands for the term number.

Common misconception

Pupils may round the wrong way when finding the term number for a particular term value.

By calculating the terms either side of the non-integer value of $$n$$, pupils can easily determine what the rounded value of $$n$$ should be.

Pupils can move from calculating the terms either side of the $$n$$ value by considering whether the sequence is increasing or decreasing and comparing this to whether they are looking for a value greater than or less than. This reasoning approach will be more appealing to some pupils.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.

$$6n-7$$ is the __________ of the arithmetic sequence starting $$-1, 5, 11, 17, 23, ...$$

Correct answer: position-to-term rule

term-to-term rule

Correct answer: $$n^\text{th}$$ term rule

term value

Q2.

What is the $$n^\text{th}$$ term rule of the sequence $$14,22,30,38,46, ...$$?

$$n+14$$

$$n+8$$

$$14+8n$$

$$14n+8$$

Correct answer: $$8n+6$$

Q3.

What is the $$n^\text{th}$$ term rule of the sequence $$197,191,185,179,173, ...$$?

$$n-6$$

$$197n-6$$

$$197-6n$$

Correct answer: $$203-6n$$

$$203n-6$$

Q4.

What is the $$40^\text{th}$$ term of the sequence $$9n+51$$?

Correct Answer: 411, n=411

Q5.

What position is $$213$$ in the following arithmetic sequence? $$18,23,28,33, ...$$

Correct Answer: 40, 40th, n=40

Q6.

Solve the equation $$8n-37=495$$ If necessary leave your answer as a decimal. $$n=$$

Correct Answer: 66.5, n=66.5

Exit quiz

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6 Questions

Q1.

$$5n-28=100$$ is __________ we can solve in order to help us find the first term over $$100$$

an expression

a solution

Correct answer: an equation

a term-to-term rule

Q2.

Using trial and error, which is the best estimate at which to start to find the first value over $$1000$$ in the sequence starting $$-17,-13,-9,-5,-1, ...$$?

$$n=10$$

$$n=100$$

$$n=1000$$

$$n=25$$

Correct answer: $$n=250$$

Q3.

If you open a savings account with $$£50$$ and put $$£8$$ a week in to the account every week thereafter without withdrawing anything, after how many weeks will you reach $$£200$$?

4 weeks

Correct answer: 19 weeks

20 weeks

25 weeks

Q4.

Which term is the first value below $$−300$$ in the arithmetic sequence starting $$91, 83, 75, 67, 59, …$$?

The $$47^\text{th}$$ term

The $$48^\text{th}$$ term

The $$49^\text{th}$$ term

Correct answer: The $$50^\text{th}$$ term

The $$51^\text{st}$$ term

Q5.

The solution to $$2n-19=98$$ is $$n=58.5$$ What does this tell us about this sequence? $$-17,-15,-13,-11, ...$$

The $$58.5^\text{th}$$ term is $$98$$

The $$58^\text{th}$$ term is above $$98$$

Correct answer: The $$59^\text{th}$$ term is above $$98$$

Q6.

What is the last positive value in the sequence $$380, 365, 350, 335, ...$$?

$$n=26$$

Correct answer: $$5$$

$$n=27$$

$$10$$