🔢 Sequences & Patterns

Grade 5 · Algebra · Cambridge Primary Stage 5

What is a Sequence?

Terms, positions, patterns

Finding the Rule

Look at the differences

The nth Term

4n − 1 formula

Geometric Sequences

Multiply by the same number

Special Sequences

Square, triangular, Fibonacci

1. What is a Sequence?

A sequence is a list of numbers that follow a rule. Each number in the list is called a term.

3, 7, 11, 15, 19, ...
The 1st term is 3, the 2nd term is 7, the 3rd term is 11, and so on.
The position of a term is its place in the list (1st, 2nd, 3rd...).
An arithmetic (linear) sequence adds or subtracts the same number each time.
💡 The "..." at the end means the sequence continues — it does not stop there!

2. Finding the Rule

To find the rule of an arithmetic sequence, look at the difference between consecutive terms.

Sequence: 5, 12, 19, 26, 33
Differences: 7, 7, 7, 7
Rule: start at 5, add 7 each time
Sequence: 30, 24, 18, 12, 6
Differences: −6, −6, −6, −6
Rule: start at 30, subtract 6 each time
💡 Always check your rule works by generating the next term and seeing if it fits the pattern.

3. The nth Term of a Linear Sequence

The nth term formula lets you find any term in the sequence without writing them all out.

nth term = (common difference) × n + (adjustment)
Sequence: 3, 7, 11, 15 — common difference = 4
nth term = 4n + ?  Check n=1: 4×1 = 4, but we want 3, so adjustment = −1
nth term = 4n − 1
Verify: n=1 → 3 ✓   n=2 → 7 ✓   n=3 → 11 ✓
Sequence: 2, 5, 8, 11 — difference = 3
nth term = 3n + ?  n=1: 3×1 = 3, want 2, so − 1
nth term = 3n − 1

4. Geometric Sequences

In a geometric sequence, you multiply by the same number each time. This is called the common ratio.

2, 6, 18, 54, 162 — multiply by 3 each time (ratio = 3)
5, 10, 20, 40, 80 — multiply by 2 each time (doubling)
64, 32, 16, 8, 4 — multiply by 1/2 each time (halving)
💡 To spot a geometric sequence, check if dividing consecutive terms always gives the same number (the ratio).

5. Special Sequences

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (n²)
Differences: 3, 5, 7, 9, 11 (odd numbers!)
Triangular numbers: 1, 3, 6, 10, 15, 21, 28 (add 1 more each time: +2, +3, +4...)
These are the number of dots you can arrange in triangles.
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Rule: each term = sum of the two terms before it.

Example 1 — Continue a Sequence

Sequence: 4, 9, 14, 19, ? — what comes next?
Difference = 5 (add 5 each time). 19 + 5 = 24

Example 2 — Find the Common Difference

Sequence: 7, 14, 21, 28, 35. What is the rule?
14 − 7 = 7, 21 − 14 = 7. Common difference = 7

Example 3 — Use the nth Term

nth term = 3n + 1. Find T(8).
T(8) = 3 × 8 + 1 = 24 + 1 = 25

Example 4 — Square and Triangular Numbers

What is the 7th square number?
7² = 7 × 7 = 49

Example 5 — Geometric Sequence

Sequence: 3, 6, 12, 24, ?
Multiply by 2 each time. 24 × 2 = 48

Sequence Generator

Enter the first term and common difference to generate the next 8 terms.

nth Term Calculator

Enter a and b for the formula T(n) = an + b, and n to find the term.

Exercise 1 — Continue Arithmetic Sequences

Find the next term in each sequence.

1. 3, 7, 11, 15, ? → next term

2. 5, 12, 19, 26, ? → next term

3. 30, 24, 18, 12, ? → next term

4. 2, 9, 16, 23, ? → next term

5. 100, 93, 86, 79, ? → next term

6. 4, 11, 18, 25, ? → next term

7. 50, 41, 32, 23, ? → next term

8. 6, 13, 20, 27, ? → next term

9. 1, 8, 15, 22, ? → next term

10. 20, 14, 8, 2, ? → next term

Exercise 2 — Finding the Rule

Enter the common difference (or multiplier for geometric sequences).

1. Sequence: 4, 11, 18, 25. Common difference?

2. Sequence: 30, 24, 18, 12. Common difference? (negative — enter −6)

3. Sequence: 2, 6, 18, 54. Common ratio (multiplier)?

4. Sequence: 5, 10, 15, 20. Common difference?

5. Sequence: 3, 7, 11, 15. Common difference?

6. Sequence: 64, 32, 16, 8. Common ratio (multiplier — enter 2 for halving)? Enter the value you multiply by to get next term. (0.5)

7. Sequence: 1, 4, 7, 10. Common difference?

8. Sequence: 45, 40, 35, 30. Common difference? (enter −5)

9. Sequence: 2, 10, 50, 250. Common ratio?

10. Sequence: 6, 11, 16, 21. Common difference?

Exercise 3 — Using the nth Term

Use the nth term formula to find the given term.

1. nth term = 3n + 1. Find T(8)

2. nth term = 4n − 1. Find T(5)

3. nth term = 2n + 5. Find T(10)

4. nth term = 5n. Find T(7)

5. nth term = n + 9. Find T(12)

6. nth term = 6n − 4. Find T(6)

7. nth term = 10n + 3. Find T(4)

8. nth term = 3n − 2. Find T(9)

9. nth term = 7n + 1. Find T(3)

10. nth term = 4n + 6. Find T(11)

Exercise 4 — Special Sequences

Answer questions about square numbers, triangular numbers and Fibonacci.

1. What is the 5th square number? (5² = ?)

2. What is the 9th square number?

3. What is the 6th triangular number? (1,3,6,10,15,?)

4. What is the 8th triangular number? (1,3,6,10,15,21,28,?)

5. What is the 8th term of the Fibonacci sequence? (1,1,2,3,5,8,13,?)

6. What is the 10th square number?

7. What is the 5th triangular number? (1,3,6,10,?)

8. What is the 10th Fibonacci number? (1,1,2,3,5,8,13,21,34,?)

9. What is the 7th square number?

10. What is the 4th triangular number? (1,3,6,?)

Exercise 5 — Mixed Sequences and Patterns

Mixed questions on sequences.

1. Find the missing term: 8, 15, 22, ?, 36

2. The nth term is 5n + 2. What is T(6)?

3. Sequence: 3, 9, 27, 81, ? (geometric). Find next term.

4. What is the 6th square number?

5. Sequence: 100, 94, 88, 82, ?. Find next term.

6. 7th triangular number? (1,3,6,10,15,21,?)

7. nth term = 9n − 3. Find T(7).

8. Sequence: 1, 4, 9, 16, ? (squares). Find next term.

9. Fibonacci: 1,1,2,3,5,8,13,21,? Find next term.

10. Sequence: 5, 9, 13, 17, ?. Common difference?

🏋️ Practice — Mixed Sequences (20 Questions)

1. Sequence: 6, 13, 20, 27. Next term?

2. nth term = 2n + 3. Find T(7).

3. What is the 4th square number?

4. Sequence: 80, 72, 64, 56. Next term?

5. 5th triangular number? (1,3,6,10,?)

6. nth term = 4n + 5. Find T(9).

7. Sequence: 1, 2, 4, 8, 16. Next term? (geometric)

8. 7th Fibonacci number? (1,1,2,3,5,8,?)

9. Common difference of: 15, 22, 29, 36?

10. nth term = 6n − 5. Find T(4).

11. Sequence: 3, 12, 48, 192. Next term? (geometric)

12. 8th square number?

13. Sequence: 50, 43, 36, 29. Next term?

14. nth term = 3n + 7. Find T(6).

15. 6th triangular number? (1,3,6,10,15,?)

16. Common difference of: 8, 3, −2, −7?

17. nth term = 7n. Find T(8).

18. Fibonacci: 13th term? (1,1,2,3,5,8,13,21,34,55,89,144,?)

19. Sequence: 2, 6, 18, 54. Multiplier?

20. nth term = 5n − 3. Find T(10).

🏆 Challenge — Advanced Sequence Problems (8 Questions)

1. The nth term = 4n − 1. What position is the term 39? (find n)

2. A sequence has first term 7 and common difference 5. What is the 20th term?

3. Two sequences: A: 3n+1 and B: 2n+5. What is the first value of n where A is greater than B?

4. Geometric: first term = 5, ratio = 3. Find the 5th term.

5. The sum of the 3rd and 5th triangular numbers is? (6+15=?)

6. Find the nth term of: 5, 8, 11, 14. Then find T(15).

7. The nth term = 3n² is NOT a linear sequence. What is T(4)? (3×16=?)

8. Fibonacci: add the first 8 terms (1+1+2+3+5+8+13+21=?)