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Finding Terms of a Sequence
Year 7 • Sequences
Lesson: Finding the Terms of a Sequence
Learn how to continue number patterns, find missing numbers, and work out any term in a simple sequence.
This lesson is perfect for Year 7 students who are practising sequences and preparing for higher-level algebra.
Lesson Objectives
- Understand what a sequence and a term are.
- Identify the rule of a simple increasing or decreasing sequence.
- Continue a sequence and find later terms, such as the 7th or 10th term.
- Find missing terms in a sequence using the rule.
- Use sequences to solve simple real-life problems.
1. What Is a Sequence?
A sequence is an ordered list of numbers that follows a rule.
Each number in the list is called a term of the sequence.
Example sequences:
3, 6, 9, 12, … (add 3 each time)
20, 18, 16, 14, … (subtract 2 each time)
5, 10, 20, 40, … (multiply by 2 each time)
3, 6, 9, 12, … (add 3 each time)
20, 18, 16, 14, … (subtract 2 each time)
5, 10, 20, 40, … (multiply by 2 each time)
We can talk about the 1st term, 2nd term, 3rd term, and so on.
2. Spotting the Rule
To work with a sequence, we first need to work out its rule.
Look at how the numbers change from one term to the next.
Example:
Sequence: 4, 7, 10, 13, …
Sequence: 4, 7, 10, 13, …
7 − 4 = 3
10 − 7 = 3
13 − 10 = 3
The rule is: “add 3 each time”.
So the next terms will be 16, 19, 22, …
3. Finding Later Terms in a Sequence
Once we know the rule, we can keep applying it to find later terms like the 7th term or 10th term.
Example 1:
Sequence: 2, 5, 8, 11, … (add 3 each time)
Sequence: 2, 5, 8, 11, … (add 3 each time)
Terms:
1st term = 2
2nd term = 5
3rd term = 8
4th term = 11
5th term = 14
6th term = 17
7th term = 20
So the 7th term is 20.
Example 2:
Sequence: 30, 27, 24, 21, … (subtract 3 each time)
Sequence: 30, 27, 24, 21, … (subtract 3 each time)
1st term = 30
2nd term = 27
3rd term = 24
4th term = 21
5th term = 18
6th term = 15
7th term = 12
So the 7th term is 12.
4. Finding Missing Terms
Sometimes a term in the middle of the sequence is missing.
We can still find it by using the rule and the terms we do know.
Example:
3, ☐, 11, 15, 19, …
3, ☐, 11, 15, 19, …
Look at the known terms: 11, 15, 19
15 − 11 = 4, 19 − 15 = 4
The rule is “add 4”.
So the missing term is: 3 + 4 = 7
The full sequence is: 3, 7, 11, 15, 19, …
5. Real-Life Sequence Example
A student saves £5 in Week 1, and then adds £3 more every week.
The amount saved forms a sequence:
Week 1: £5
Week 2: £8
Week 3: £11
Week 4: £14, and so on.
(a) How much money will the student have in Week 6?
(b) In which week will the student first reach at least £26?
(Answers are in the Answer Key section below.)
Practice Questions
A. Continue the Sequences
Write the next three terms in each sequence.
- 2, 5, 8, 11, …
- 30, 27, 24, 21, …
- 10, 20, 30, 40, …
B. Rule and Later Terms
For each sequence:
- State whether it is increasing or decreasing.
- Write the rule in words (for example: “add 4 each time”).
- Find the term asked for.
- 4, 7, 10, 13, … Find the 7th term.
- 50, 45, 40, 35, … Find the 9th term.
- 6, 9, 12, 15, … Which term is 39?
C. Find the Missing Terms
Fill in the missing terms and state the rule of each sequence.
- 3, ☐, 11, 15, 19, …
- 25, 20, ☐, 10, 5, …
- 5, ☐, 13, 17, 21, …
- 10, ☐, ☐, 22, 26, …
D. Word Problems with Sequences
-
A step counter shows 2,000 steps on Monday, and increases by 500 steps each day.
(a) How many steps on Thursday?
(b) On which day will the steps first reach 4,000? -
A pattern of tiles goes: 4 tiles in the 1st row, 7 tiles in the 2nd row, 10 tiles in the 3rd row, and so on.
(a) How many tiles are in the 5th row?
(b) How many tiles are in the 8th row?
E. Challenge – Does This Number Appear?
A sequence starts at 2 and each time we add 5:
2, 7, 12, 17, 22, …
- Write down the 6th and 7th terms.
- Will the number 102 appear in this sequence? Explain your reasoning.
✅ Show Answer Key
A. Continue the Sequences
- 2, 5, 8, 11, 14, 17, 20
- 30, 27, 24, 21, 18, 15, 12
- 10, 20, 30, 40, 50, 60, 70
B. Rule and Later Terms
-
4, 7, 10, 13, …
Increasing, rule: add 3 each time.
Terms: 4, 7, 10, 13, 16, 19, 22 → 7th term = 22. -
50, 45, 40, 35, …
Decreasing, rule: subtract 5 each time.
Terms: 50, 45, 40, 35, 30, 25, 20, 15, 10 → 9th term = 10. -
6, 9, 12, 15, …
Increasing, rule: add 3 each time.
Terms: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39.
So 39 is the 12th term.
C. Find the Missing Terms
- 3, ☐, 11, 15, 19, …
Rule: add 4. Missing term = 7. - 25, 20, ☐, 10, 5, …
Rule: subtract 5. Missing term = 15. - 5, ☐, 13, 17, 21, …
Rule: add 4. Sequence: 5, 9, 13, 17, 21 → missing term = 9. - 10, ☐, ☐, 22, 26, …
Rule: add 4. Sequence: 10, 14, 18, 22, 26 → missing terms = 14 and 18.
D. Word Problems with Sequences
-
Steps: start at 2,000 and add 500 each day.
Monday: 2,000 (Day 1)
Tuesday: 2,500 (Day 2)
Wednesday: 3,000 (Day 3)
Thursday: 3,500 (Day 4)
Friday: 4,000 (Day 5)(a) Thursday = 3,500 steps.
(b) 4,000 steps on Friday. -
Tiles: 4, 7, 10, 13, … (add 3 each time).
1st row: 4
2nd row: 7
3rd row: 10
4th row: 13
5th row: 16 tilesContinue: 6th: 19, 7th: 22, 8th: 25 tiles.
(a) 5th row = 16 tiles.
(b) 8th row = 25 tiles.
Real-Life Example (Section 5)
-
Savings: start at £5, add £3 each week.
Week 1: 5
Week 2: 8
Week 3: 11
Week 4: 14
Week 5: 17
Week 6: 20
So in Week 6 the student has £20. -
Continue: Week 7: 23, Week 8: 26, Week 9: 29, …
The amount first reaches at least £26 in Week 8.
E. Challenge – Does 102 Appear?
-
Sequence: 2, 7, 12, 17, 22, … (add 5 each time).
6th term = 27, 7th term = 32. -
We start at 2 and keep adding 5:
2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102, …After some steps we reach 102, so yes, 102 does appear in the sequence.
© Aviate Learning – Sequences: Finding the Terms (Year 7)