🔢 Sequence Spotter 🌟

A sequence is a list of numbers that follow a rule. Being able to spot the rule, extend sequences and find missing terms is a key skill in Grade 6. You'll work with arithmetic sequences (add/subtract a fixed amount), geometric sequences (multiply/divide by a fixed amount), and special sequences like square numbers and triangular numbers.

➕ Arithmetic Sequence
3, 7, 11, 15, ... → add 4 each time
(common difference = +4)
✖️ Geometric Sequence
2, 6, 18, 54, ... → multiply by 3 each time
(common ratio = ×3)
⬛ Square Numbers
1, 4, 9, 16, 25, ... → 1², 2², 3², 4², 5², ...

📏 Golden Rules

💡 Quick Check: What type of sequence is 5, 10, 20, 40, 80?

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📖 Worked Examples

Example 1: Arithmetic Sequence

Find the next 3 terms and the 20th term of: 5, 12, 19, 26, ...

🔍 Common difference: d = 12 − 5 = 7
➕ Next 3 terms: 26 + 7 = 33, 33 + 7 = 40, 40 + 7 = 47
📐 nth term rule: first term + (n−1) × d = 5 + (n−1) × 7 = 7n − 2
🎯 20th term: 7 × 20 − 2 = 140 − 2 = 138

Example 2: Geometric Sequence

Find the next 3 terms of: 4, 12, 36, 108, ...

🔍 Common ratio: r = 12 ÷ 4 = 3
✖️ Next terms: 108 × 3 = 324, 324 × 3 = 972, 972 × 3 = 2916

Example 3: Find the Missing Term

Find the missing term: 8, __, 32, 64, 128

🔍 Check the ratio: 64 ÷ 32 = 2, so common ratio = ×2
✅ Verify: 32 ÷ 2 = 16 ✓ and 16 × 2 = 32 ✓ and 8 × 2 = 16 ✓
🎯 Missing term: 8 × 2 = 16

Example 4: Special Sequences

Is 49 a square number? What are the first 6 triangular numbers?

⬛ Is 49 a square? 49 = 7², so yes, 49 is a square number. √49 = 7.
🔺 Triangular numbers: 1, 3, 6, 10, 15, 21
➕ Differences: +2, +3, +4, +5, +6 (each gap increases by 1)
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🔬 Sequence Generator

Build your own sequence and explore how it grows!

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🎯 Drag 1: Arithmetic Next Term

The sequence is 6, 11, 16, 21, ... What comes next?

25
26
27
28
6 → 11 → 16 → 21 →
?

Hint: Common difference = 5. 21 + 5 = ?

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🎯 Drag 2: Geometric Next Term

The sequence is 3, 9, 27, 81, ... What comes next?

162
243
324
108
3 → 9 → 27 → 81 →
?

Hint: Common ratio = 3. 81 × 3 = ?

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🎯 Drag 3: Missing Term

Find the missing term: 50, 43, __, 29, 22

34
36
35
38
50 → 43 →
?
→ 29 → 22

Hint: Common difference = −7. 43 − 7 = ?

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🎯 Drag 4: nth Term

The rule for a sequence is: nth term = 4n + 1. What is the 6th term?

24
25
26
27
4n + 1, where n = 6 →
?

Hint: Substitute n = 6. 4 × 6 + 1 = ?

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🎯 Drag 5: Square Numbers

Which of these is NOT a square number? Drag it to the drop zone.

16
36
45
49
The non-square number is →
?

Hint: Check: 4²=16, 6²=36, 7²=49. What about 45?

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🎯 Drag 6: Fibonacci-type

Continue this Fibonacci-type sequence: 2, 3, 5, 8, 13, ...

18
19
20
21
2, 3, 5, 8, 13, →
?

Hint: Add the last two terms. 8 + 13 = ?

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✏️ Practice Questions

Try all 20 questions before revealing the answers!

  1. Write the next 3 terms: 4, 10, 16, 22, ...
  2. Write the next 3 terms: 100, 93, 86, 79, ...
  3. Find the common difference: 7, 15, 23, 31
  4. Find the common ratio: 2, 8, 32, 128
  5. Find the missing term: 5, __, 35, 65
  6. Find the missing term: 3, 9, __, 81, 243
  7. What is the 10th term of the sequence: 1, 5, 9, 13, ...?
  8. What is the 8th term of the sequence with nth term = 3n − 1?
  9. Write the first 5 square numbers.
  10. Write the first 5 triangular numbers.
  11. Is 64 a square number? If so, what is its square root?
  12. Is 50 a square number? Explain your answer.
  13. Find the next term: 1, 2, 4, 8, 16, ...
  14. Find the next term: 1, 1, 2, 3, 5, 8, ...
  15. The nth term of a sequence is 5n + 2. Find the 1st, 5th and 10th terms.
  16. A sequence starts at 200 and decreases by 15 each time. Write the first 5 terms.
  17. Find the two missing terms: 3, __, __, 24, 48
  18. What type of sequence is 1, 3, 6, 10, 15?
  19. The 5th term of an arithmetic sequence is 31 and the common difference is 6. Find the 1st term.
  20. Write the first 5 terms of the sequence with nth term = n² + 1.

Answers

  1. 28, 34, 40 (d = +6)
  2. 72, 65, 58 (d = −7)
  3. Common difference = +8
  4. Common ratio = ×4
  5. 20 (d = 15; the sequence is 5, 20, 35, 50, 65)
  6. 27 (common ratio = ×3)
  7. 37 (a = 1, d = 4; 10th term = 1 + 9×4 = 37)
  8. 23 (3×8 − 1 = 24 − 1 = 23)
  9. 1, 4, 9, 16, 25
  10. 1, 3, 6, 10, 15
  11. Yes, 64 is a square number. √64 = 8
  12. No — 7² = 49 and 8² = 64, so 50 is between two consecutive square numbers.
  13. 32 (common ratio = ×2)
  14. 13 (Fibonacci: 5 + 8 = 13)
  15. 1st term: 5×1+2 = 7; 5th term: 5×5+2 = 27; 10th term: 5×10+2 = 52
  16. 200, 185, 170, 155, 140
  17. 6, 12 (common ratio = ×2)
  18. Triangular numbers
  19. 7 (5th term = 1st term + 4×6 = 31, so 1st term = 31 − 24 = 7)
  20. 2, 5, 10, 17, 26 (1²+1, 2²+1, 3²+1, 4²+1, 5²+1)
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🏆 Challenge Problems

These questions require deeper thinking — show all your working!

  1. A sequence of even numbers starts at 2. Write the nth term rule and find the 50th term.
  2. Bacteria double every hour. Starting with 5 bacteria, how many are there after 6 hours?
  3. A staircase has 1 step in the 1st section, 3 in the 2nd, 6 in the 3rd, 10 in the 4th (triangular numbers). How many steps are in the 8th section?
  4. The nth term of a sequence is 4n − 3. Is 57 a term in this sequence? Show your working.
  5. A sequence is 2, 5, 8, 11, ... What is the first term greater than 100?
  6. Squares are arranged in an L-shape: pattern 1 has 3 squares, pattern 2 has 5, pattern 3 has 7. Find the nth term and the number of squares in pattern 20.
  7. A geometric sequence has first term 5 and common ratio 2. Which term equals 160?
  8. The sum of three consecutive terms in an arithmetic sequence is 36 and the common difference is 4. Find the three terms.
  9. The sequence 1, 4, 9, 16, 25 ... (square numbers). Between which two consecutive square numbers does 80 lie?
  10. A savings sequence: week 1 save £3, week 2 save £5, week 3 save £7 (arithmetic, d=2). How much total is saved over 12 weeks? (Hint: find each week's amount and add them.)

Challenge Answers

  1. nth term = 2n; 50th term = 2 × 50 = 100
  2. 5 × 2⁶ = 5 × 64 = 320 bacteria
  3. The 8th triangular number: 8 × (8+1) ÷ 2 = 8 × 9 ÷ 2 = 36 steps
  4. 4n − 3 = 57 → 4n = 60 → n = 15. Yes, 57 is the 15th term.
  5. nth term = 3n − 1. Solve 3n − 1 > 100 → n > 33.67 → n = 34. 34th term = 3×34 − 1 = 101
  6. nth term = 2n + 1. Pattern 20 = 2×20 + 1 = 41 squares
  7. 5 × 2^(n−1) = 160 → 2^(n−1) = 32 = 2⁵ → n − 1 = 5 → n = 6. It is the 6th term.
  8. Middle term = 36 ÷ 3 = 12. Terms are: 12 − 4 = 8, 12, 12 + 4 = 16 → 8, 12, 16
  9. 8² = 64 and 9² = 81. 80 lies between 64 and 81. (√80 ≈ 8.9)
  10. Week 1 = £3, Week 12 = 3 + 11×2 = £25. Sum = 12 × (3+25) ÷ 2 = 12 × 14 = £168
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