Mathematics • Year 8 • Unit 2 • Lesson 3
Number Patterns and Rules
Build fluency in two skills: finding the term-to-term rule of a sequence, and using a position-to-term rule to predict any term. One worked example, one guided example with blanks, then eight independent problems from continuing patterns to finding the 100th term.
1. I do — fully worked example
Read every line. Each step explains why, so you can copy the same thinking.
Problem. A sequence starts 2, 5, 8, 11, 14, … Find the term-to-term rule, the position-to-term rule, and use it to find the 10th term.
Step 1 — Find the term-to-term rule (what happens each time).
5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3.
Reason: every consecutive pair has the same difference, +3. So the term-to-term rule is "add 3".
Step 2 — Connect each term to its position using a table.
n: 1 2 3 4 5
term: 2 5 8 11 14
Reason: linking position number to term value reveals the formula.
Step 3 — Guess and test the formula.
Common difference is 3, so try 3n. 3 × 1 = 3, but the 1st term is 2 → need to subtract 1.
Try term = 3n − 1. Check: n = 2 → 3(2) − 1 = 5 ✓. n = 5 → 3(5) − 1 = 14 ✓.
Reason: the formula must work for every position, not just one.
Step 4 — Use the formula to predict the 10th term.
term = 3n − 1, n = 10 → 3(10) − 1 = 30 − 1 = 29.
Reason: with the rule we can skip straight to any term — no listing needed.
Answer: term-to-term +3, position-to-term 3n − 1, 10th term = 29.
2. We do — fill in the missing steps
Same shape as Section 1. Fill every blank. 4 marks
Problem. A sequence starts 4, 7, 10, 13, 16, … Find the term-to-term rule, the position-to-term rule, then find the 20th term.
Step 1 — Term-to-term:
7 − 4 = ____, 10 − 7 = ____, 13 − 10 = ____. Rule: "add ____".
Step 2 — Table:
n: 1 2 3 4 5
term: 4 ___ ___ ___ ___
Step 3 — Guess and test: common difference ____ → try ____n.
For n = 1, ____n = ____, but term 1 is 4 → add ____.
Formula: term = ____n + ____. Check n = 3: __________ = ____ ✓
Step 4 — 20th term:
term = ____(20) + ____ = ____
3. You do — independent practice
Show your working. 3.1–3.4 are foundation (continue a pattern / find the next two). 3.5–3.6 are standard (use a given position-to-term rule). 3.7–3.8 are extension (build the rule from scratch and predict a far term).
Foundation — continue the pattern
3.1 Write the next two terms of: 6, 11, 16, 21, ___, ___. State the term-to-term rule. 1 mark
3.2 Write the next two terms of: 50, 45, 40, 35, ___, ___. State the term-to-term rule. 1 mark
3.3 Write the next two terms of: 3, 6, 12, 24, ___, ___. State the term-to-term rule (not all rules are "add"). 1 mark
3.4 Write the next two terms of: 1, 4, 9, 16, ___, ___. (Hint: these are the square numbers.) State a position-to-term rule. 1 mark
Standard — use a position-to-term rule
3.5 A sequence has position-to-term rule term = 2n + 5. Find the 1st, 2nd, 3rd and 8th terms. 2 marks
3.6 A sequence has position-to-term rule term = 4n − 3. Which term equals 33? (Solve 4n − 3 = 33.) 2 marks
Extension — build the rule and predict
3.7 A sequence starts 5, 8, 11, 14, 17, … (a) Find the position-to-term rule. (b) Use it to find the 100th term. 2 marks
3.8 A sequence starts 1, 3, 5, 7, 9, … (the odd numbers). (a) Find the position-to-term rule. (b) Will 250 ever appear in this sequence? Justify briefly. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do 4, 7, 10, 13, 16
Step 1: differences 3, 3, 3. Rule: "add 3".
Step 2: terms 4, 7, 10, 13, 16.
Step 3: common difference 3 → try 3n. For n = 1, 3n = 3, but term 1 is 4 → add 1. Formula: term = 3n + 1. Check n = 3: 3(3) + 1 = 10 ✓.
Step 4: 3(20) + 1 = 61. So the 20th term is 61.
3.1 — 6, 11, 16, 21
Next two: 26, 31. Rule: "add 5".
3.2 — 50, 45, 40, 35
Next two: 30, 25. Rule: "subtract 5".
3.3 — 3, 6, 12, 24
Each term is double the previous one. Next two: 48, 96. Rule: "multiply by 2".
3.4 — 1, 4, 9, 16
Next two: 25, 36. Position-to-term rule: term = n² (square numbers).
3.5 — Rule term = 2n + 5
n = 1: 2(1) + 5 = 7. n = 2: 9. n = 3: 11. n = 8: 2(8) + 5 = 21.
3.6 — When does 4n − 3 = 33?
4n − 3 = 33 → 4n = 36 → n = 9. So the 9th term equals 33. Check: 4(9) − 3 = 33 ✓.
3.7 — 5, 8, 11, 14, 17, …
(a) Common difference 3 → try 3n. For n = 1, 3n = 3, but term 1 is 5 → add 2. Rule: term = 3n + 2.
(b) 100th term: 3(100) + 2 = 302.
3.8 — Odd numbers 1, 3, 5, 7, 9, …
(a) Common difference 2 → try 2n. For n = 1, 2n = 2, but term 1 is 1 → subtract 1. Rule: term = 2n − 1.
(b) No, 250 will never appear. Setting 2n − 1 = 250 gives 2n = 251 → n = 125.5, which is not a whole number. Also, every term of 2n − 1 is odd (even − 1 = odd), and 250 is even.