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Year 7 Maths – Understanding Sequences | Term-to-Term Rule Explained
Year 7 • Sequences
Lesson: Sequences
Learn how to understand and work with number sequences: recognise patterns, find the next terms, describe the term-to-term rule, and identify increasing and decreasing sequences – perfect for Year 7 students.
Lesson Objectives
- Understand what a sequence and a term are.
- Recognise simple number patterns in a sequence.
- Find the next terms in a sequence using the rule.
- Describe and write the term-to-term rule in words and using symbols.
- Identify whether a sequence is increasing or decreasing.
1. What Is a Sequence?
A sequence is an ordered list of numbers that follow a pattern or rule.
Each number in the list is called a term.
Examples of simple sequences:
2, 4, 6, 8, 10, … (add 2 each time)
10, 8, 6, 4, 2, … (subtract 2 each time)
3, 6, 9, 12, … (add 3 each time)
2, 4, 6, 8, 10, … (add 2 each time)
10, 8, 6, 4, 2, … (subtract 2 each time)
3, 6, 9, 12, … (add 3 each time)
2. Term-to-Term Rule
The term-to-term rule tells you how to get from one term in the sequence to the next.
Usually, we add, subtract, multiply or divide by the same number each time.
Example 1:
Sequence: 5, 8, 11, 14, …
5 → 8: +3, 8 → 11: +3, 11 → 14: +3
Sequence: 5, 8, 11, 14, …
5 → 8: +3, 8 → 11: +3, 11 → 14: +3
Term-to-term rule: add 3 each time.
Example 2:
Sequence: 20, 15, 10, 5, …
20 → 15: −5, 15 → 10: −5, 10 → 5: −5
Term-to-term rule: subtract 5 each time.
3. Finding the Next Terms in a Sequence
Once you know the rule, you can continue the sequence by applying the rule again and again.
Example:
Sequence: 4, 7, 10, 13, …
Sequence: 4, 7, 10, 13, …
Step 1: Find the difference between terms: 7 − 4 = 3, 10 − 7 = 3, 13 − 10 = 3
Step 2: So the rule is add 3 each time.
Next terms:
13 + 3 = 16
16 + 3 = 19
19 + 3 = 22
So the sequence continues as: 4, 7, 10, 13, 16, 19, 22, …
4. Increasing and Decreasing Sequences
We can describe sequences as increasing or decreasing:
Increasing sequence: The terms get bigger each time (for example, add 2 each step).
Decreasing sequence: The terms get smaller each time (for example, subtract 4 each step).
Decreasing sequence: The terms get smaller each time (for example, subtract 4 each step).
Example – Increasing:
1, 4, 7, 10, … → add 3 each time → increasing sequence.
Example – Decreasing:
30, 25, 20, 15, … → subtract 5 each time → decreasing sequence.
5. Pattern Example with a Story
A student saves money each week. On week 1 they save £5, on week 2 they save £8, on week 3 they save £11, and so on.
The sequence of savings is: 5, 8, 11, 14, …
(a) What is the term-to-term rule?
(b) How much will they save in week 5?
(Answers are in the Answer Key section below.)
Practice Questions
A. Find the Next Terms
Write the next three terms in each sequence.
- 2, 5, 8, 11, …
- 10, 7, 4, 1, …
- 6, 9, 12, 15, …
B. Describe the Rule
For each sequence, write the term-to-term rule in words.
- 4, 9, 14, 19, …
- 25, 21, 17, 13, …
- 3, 6, 12, 24, …
C. Increasing or Decreasing?
State whether each sequence is increasing or decreasing.
- 7, 10, 13, 16, …
- 50, 45, 40, 35, …
- 2, 4, 8, 16, …
D. Mixed Practice
- The first term of a sequence is 3 and the term-to-term rule is “add 4”.
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