5.8 Using the Nth Term of a Sequence

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: generating terms from the nth term, finding the nth term of an arithmetic sequence and using it. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Use an nth term rule to generate terms of a sequence
• Find the nth term of an arithmetic (linear) sequence
• Decide whether a number is a term of a sequence
• Use the nth term to solve problems about patterns

Key words

• Position: Where a term is in the sequence: 1st, 2nd, 3rd, … This is n.
• nth term: A rule that gives any term from its position, e.g. 3n + 2.
• Position-to- term rule: Another name for the nth term.
• Common difference: The amount added each time in an arithmetic sequence.
• Linear sequence: Another name for an arithmetic sequence.

Generating terms from the nth term

Substitute n = 1, 2, 3, … into the rule. To find the 20th term, substitute n = 20. You do not need to list all the terms before it.

• 3n + 2; 1 5; 2 8; 3 11; 4 14; 20 62
• 50 − 4n; 1 46; 2 42; 3 38; 4 34; 20 −30
• n² + 1; 1 2; 2 5; 3 10; 4 17; 20 401

Worked example: the 20th term

Find the 20th term of the sequence with nth term 3n + 2.
3 × 20 + 2 = 60 + 2 = 62

Answer: 62

Finding the nth term

For an arithmetic sequence the nth term has the form dn + c, where d is the common difference.

• Step 1: find the common difference d. The rule starts dn.
• Step 2: write out the d times table under the sequence.
• Step 3: what do you add or subtract to get from the times table to the sequence?

Worked example: an increasing sequence

Find the nth term of 7, 11, 15, 19, …
Difference = +4, so start with 4n: 4, 8, 12, 16
Each term is 3 more than 4n.

Answer: 4n + 3

Worked example: a decreasing sequence

Find the nth term of 20, 17, 14, 11, …
Difference = −3, so start with −3n: −3, −6, −9, −12
Each term is 23 more than −3n.

Answer: 23 − 3n (or −3n + 23)

Key fact: Check your rule with n = 1: for 4n + 3, 4 × 1 + 3 = 7 ✓. It should give the first term.

Is a number in the sequence?

Set the nth term equal to the number and solve. If n is a positive whole number, the number is in the sequence.

Worked example: testing two numbers

The nth term of a sequence is 4n + 3. Are 99 and 101 terms of the sequence?
4n + 3 = 99 gives 4n = 96, n = 24. Whole number: 99 is the 24th term.
4n + 3 = 101 gives 4n = 98, n = 24.5. Not whole: 101 is not a term.

Answer: 99 is a term, 101 is not

Patterns and problems

Worked example: using the nth term of a pattern

(a) How many sticks are in pattern 50? 3 × 50 + 1 = 151
(b) Which pattern uses 91 sticks? 3n + 1 = 91, 3n = 90, n = 30

Answer: (a) 151 sticks (b) pattern 30

Worked example: first term over a value

Which is the first term of 6n − 5 that is greater than 100?
6n − 5 > 100, 6n > 105, n > 17.5. So n = 18: 6 × 18 − 5 = 103.

Answer: 103 (the 18th term)

Common mistakes

• Using the first term as the number in front of n. For 7, 11, 15, … it is 4n + 3, not 7n + 4.
• Writing the rule as n + 4 for 6, 10, 14, … The term-to-term rule (+4) is not the nth term.
• Forgetting the negative for a decreasing sequence: 20, 17, 14, … is 23 − 3n.
• Saying a number is in a sequence when n is not a whole number.

Revision questions

Try these, then check your answers below each one.

Question 1. The nth term of a sequence is 4n - 1. Work out the 15th term.

Answer: 59

Working: 4 × 15 - 1 = 59

Question 2. Find the nth term of 3, 8, 13, 18, … (Choose from: 5n - 2, 5n + 3, n + 5, 3n + 5.)

Answer: 5n - 2

Working: 5n: 5, 10, 15, … Subtract 2 to get 3, 8, 13, …: 5n - 2.

Question 3. The nth term of 9, 13, 17, 21, … is 4n + b. What is the value of b?

Answer: 5

Working: 9 - 4 = 5, so the nth term is 4n + 5.

Related lessons

• Previous lesson: 5.7 Generating Sequences
• Next lesson: 6.1 Properties of Shapes
• In this chapter: 5.1 Solving Equations 1
• In this chapter: 5.2 Solving Equations 2
• In this chapter: 5.3 Solving Equations with Brackets

Continue in the full course

This lesson is section 5.8 (EGCMF0508) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.