5.7 Generating Sequences

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: term-to-term rules, arithmetic and geometric sequences, special sequences and patterns. 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

• Find and use a term-to-term rule to continue a sequence
• Recognise arithmetic, geometric and Fibonacci-type sequences
• Recognise square, cube and triangular numbers
• Continue sequences of patterns and explain how they grow

Key words

• Sequence: A list of numbers or patterns that follows a rule.
• Term: One number (or pattern) in a sequence. The first term is the 1st number in the list.
• Term-to-term rule: How to get from one term to the next, e.g. “add 4”.
• Arithmetic sequence: The terms go up or down by the same amount each time (a common difference).
• Geometric sequence: Each term is multiplied by the same number to get the next.

Term-to-term rules

Look at how you get from one term to the next. Check that the same rule works all the way along.

• 5, 9, 13, 17, …; Rule add 4; Next two terms 21, 25
• 30, 24, 18, 12, …; Rule subtract 6; Next two terms 6, 0
• 7, 4, 1, −2, …; Rule subtract 3; Next two terms −5, −8
• 3, 6, 12, 24, …; Rule multiply by 2; Next two terms 48, 96
• 80, 40, 20, 10, …; Rule divide by 2; Next two terms 5, 2.5

Worked example: a two-step rule

The first term of a sequence is 6. The rule is “multiply by 3, then subtract 4”. Write the first four terms.
6 × 3 − 4 = 14, 14 × 3 − 4 = 38, 38 × 3 − 4 = 110

Answer: 6, 14, 38, 110

Types of sequence

• Arithmetic: add or subtract the same number each time, e.g. 2, 9, 16, 23 (common difference +7).
• Geometric: multiply by the same number each time, e.g. 5, 15, 45, 135 (×3).
• Fibonacci-type: add the two previous terms to get the next, e.g. 1, 1, 2, 3, 5, 8, 13.

Worked example: a Fibonacci-type sequence

The first two terms are 2 and 5. Each term is the sum of the two terms before it. Find the 6th term.
2, 5, 2 + 5 = 7, 5 + 7 = 12, 7 + 12 = 19, 12 + 19 = 31

Answer: 31

Worked example: missing terms

Fill the gaps in the arithmetic sequence 8, __, __, 23.
From 8 to 23 is 15, in 3 equal steps. 15 ÷ 3 = 5.

Answer: 8, 13, 18, 23

Special sequences

• Square numbers; Start of the sequence 1, 4, 9, 16, 25, 36, …; How it is made 1 × 1, 2 × 2, 3 × 3, …
• Cube numbers; Start of the sequence 1, 8, 27, 64, 125, …; How it is made 1 × 1 × 1, 2 × 2 × 2, …
• Triangular numbers; Start of the sequence 1, 3, 6, 10, 15, 21, …; How it is made add 2, then 3, then 4, …
• Even numbers; Start of the sequence 2, 4, 6, 8, …; How it is made add 2
• Odd numbers; Start of the sequence 1, 3, 5, 7, …; How it is made add 2

Patterns

Sequences often come from patterns of shapes. Count each pattern, find the rule, and explain where the extra pieces go.

Worked example: continue a pattern

Sticks: 4, 7, 10, 13, … The rule is add 3.
Pattern 5: 16, pattern 6: 19, pattern 7: 22, pattern 8: 25.
Is there a pattern with 50 sticks? 50 − 4 = 46, and 46 is not a multiple of 3, so no.

Answer: pattern 8 uses 25 sticks; no pattern uses 50

Common mistakes

• Checking the rule with only the first two terms. Make sure it works for every pair.
• Getting negative terms wrong: after −2, subtracting 3 gives −5, not −1.
• Mixing up arithmetic (add the same amount) and geometric (multiply by the same amount).
• Thinking 1, 3, 6, 10 has no rule: the differences 2, 3, 4 grow by 1 each time.

Revision questions

Try these, then check your answers below each one.

Question 1. Write down the next term of the sequence 7, 13, 19, 25, …

Answer: 31

Working: The rule is add 6: 25 + 6 = 31.

Question 2. Write down the next term of the sequence 96, 48, 24, 12, …

Answer: 6

Working: Divide by 2: 12 ÷ 2 = 6.

Question 3. Which of these sequences is geometric? (Choose from: 2, 6, 18, 54, …, 2, 6, 10, 14, …, 1, 4, 9, 16, …, 1, 1, 2, 3, ….)

Answer: 2, 6, 18, 54, …

Working: 2, 6, 18, 54 multiplies by 3 each time. The others add 4, are square numbers, or are Fibonacci.

Related lessons

• Previous lesson: 5.6 Using Formulae
• Next lesson: 5.8 Using the Nth Term of a Sequence
• 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 5.7 (EGCMF0507) 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.