Patterns & Sequences — Grade 4 Mathematics
In Grade 4, we move beyond just describing patterns — we find the nth term rule for arithmetic sequences, use two-step function machines, and begin describing sequences using algebraic notation.
Arithmetic Sequences and the nth Term
In an arithmetic sequence, the same number is added each time — this is the common difference (d).
The nth term formula:
Example: 5, 8, 11, 14, … → d = 3 → nth term = 5 + (n−1) × 3 = 3n + 2
The nth term formula:
nth term = first term + (n − 1) × dExample: 5, 8, 11, 14, … → d = 3 → nth term = 5 + (n−1) × 3 = 3n + 2
5
+3
8
+3
11
+3
14
+3
17
Sequence: 5, 8, 11, 14, 17, …
d = 3 → nth term = 3n + 2
Check: n=1 → 3(1)+2 = 5 ✓ n=4 → 3(4)+2 = 14 ✓
d = 3 → nth term = 3n + 2
Check: n=1 → 3(1)+2 = 5 ✓ n=4 → 3(4)+2 = 14 ✓
💡 Shortcut: nth term = d × n + (first term − d). For d=3, 1st term=5: 3n + (5−3) = 3n + 2.
Two-Step Function Machines
A function machine performs operations in order. To reverse it (find input from output), use inverse operations in reverse order.
Input
n
n
→
× 4
→
− 3
→
Output
4n − 3
4n − 3
Worked Examples
Example 1 — Continue the Sequence
Find the next two terms of 7, 12, 17, 22, …
- Common difference: 12 − 7 = 5
- 22 + 5 = 27 27 + 5 = 32
- Next terms: 27, 32
Example 2 — Find the 10th Term
Sequence: 4, 9, 14, 19, … Find the 10th term.
- d = 5; first term = 4
- nth term = 5n + (4 − 5) = 5n − 1
- 10th term = 5(10) − 1 = 50 − 1 = 49
- Check by listing: 4, 9, 14, 19, 24, 29, 34, 39, 44, 49 ✓
Example 3 — Reverse a Function Machine
A machine does × 3 then + 5. The output is 23. Find the input.
- Reverse: output 23 → undo +5: 23 − 5 = 18
- Undo × 3: 18 ÷ 3 = 6
- Check: 6 × 3 + 5 = 23 ✓
Practice Questions
1. Find the next two terms of 3, 9, 15, 21, …
✅ d = 6; 21 + 6 = 27; 27 + 6 = 27, 33
2. Write the nth term rule for 6, 11, 16, 21, …
✅ d = 5; nth term = 5n + (6−5) = 5n + 1 (check: n=1 → 6 ✓)
3. Using rule 5n + 1, find the 8th term.
✅ 5(8) + 1 = 40 + 1 = 41
4. A machine does × 6 then − 4. Input = 7. Find the output.
✅ 7 × 6 = 42; 42 − 4 = 38
5. A machine does × 5 then + 2. The output is 47. Find the input.
✅ 47 − 2 = 45; 45 ÷ 5 = 9
Key Points to Remember
- The common difference (d) is what you add each time in an arithmetic sequence.
- nth term = d × n + (1st term − d).
- Always check your nth term rule with the first term (n=1).
- To reverse a function machine: use inverse operations in reverse order.
- Inverse of × is ÷; inverse of + is −; inverse of − is +.