Mathematics • Year 8 • Unit 2 • Lesson 3
Patterns — Mixed Challenge
Pull together everything from Lesson 3: term-to-term and position-to-term rules, finding the rule from a sequence, using a rule to predict, and solving for a position. Six mixed problems, one "find the mistake", and one open-ended puzzle.
1. Mixed problems — choose the right move
Decide what's being asked BEFORE you write. Show working. 3 marks each
1.1 Find the next two terms and the term-to-term rule for: 7, 11, 15, 19, 23, ___, ___.
1.2 A sequence has position-to-term rule term = 5n − 2. Find the 1st, 4th and 12th terms.
1.3 Build a position-to-term rule for: 8, 11, 14, 17, 20, … Then use it to find the 50th term.
1.4 A sequence has rule term = 3n + 4. Which position gives the term 49? (Solve 3n + 4 = 49.)
1.5 Decide whether each sequence is increasing, decreasing, or neither (and explain in one phrase): (a) 100, 92, 84, 76, …; (b) 1, 2, 4, 8, …; (c) 5, 5, 5, 5, …
1.6 A sequence starts 20, 17, 14, 11, 8, … (a) Find the position-to-term rule. (b) Find the value of the 9th term. (c) State what the FIRST negative term in this sequence will be (i.e. its value AND its position).
2. Find the mistake
Another student tried to build a position-to-term rule for 6, 10, 14, 18, 22, … Their working is below. Exactly one line contains a mistake. Spot it, explain why, then re-do the working correctly. 3 marks
Student's working — find the rule for 6, 10, 14, 18, 22:
Line 1: Differences: 10 − 6 = 4, 14 − 10 = 4, 18 − 14 = 4, 22 − 18 = 4.
Line 2: Common difference is 4, so the rule starts with 4n.
Line 3: For n = 1, 4n = 4. But the first term is 6, so I need to ADD 6.
Line 4: So the rule is term = 4n + 6. Check n = 2: 4(2) + 6 = 14 ✓.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full and give the correct rule, then check it works for n = 1, 2 and 5.
Stuck? At n = 1 the formula must equal the first term (6). Solve 4(1) + c = 6 → c = 2, not 6.3. Open-ended challenge — design your own sequence
This question has more than one valid answer. 4 marks
3.1 Design a sequence with the following two features:
(i) The first term is 10, AND
(ii) the 5th term is 30.
For your sequence:
(a) State the term-to-term rule you chose.
(b) Write out the first five terms.
(c) Write the position-to-term rule and verify that it gives 10 for n = 1 and 30 for n = 5.
Bonus: Find a SECOND, different sequence that also satisfies the same two conditions (maybe not a constant-difference sequence).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 7, 11, 15, 19, 23
Common difference +4. Next two: 27, 31. Term-to-term rule: "add 4".
1.2 — Rule term = 5n − 2
n = 1: 3. n = 4: 5(4) − 2 = 18. n = 12: 5(12) − 2 = 58.
1.3 — 8, 11, 14, 17, 20
Common difference 3 → 3n. For n = 1, 3n = 3 but value is 8 → add 5. Rule: term = 3n + 5. 50th term: 3(50) + 5 = 155.
1.4 — When does 3n + 4 = 49?
3n + 4 = 49 → 3n = 45 → n = 15. So the 15th term equals 49.
1.5 — Increasing, decreasing, or neither?
(a) Decreasing — common difference −8.
(b) Increasing — terms get bigger (doubling each time).
(c) Neither — constant sequence; every term stays at 5.
1.6 — 20, 17, 14, 11, 8, …
(a) Common difference −3, first term 20 → rule term = 23 − 3n (equivalently −3n + 23). Check n = 1: 23 − 3 = 20 ✓.
(b) 9th term: 23 − 3(9) = 23 − 27 = −4.
(c) First negative happens when 23 − 3n < 0 → n > 23/3 ≈ 7.67 → n = 8. The 8th term is the first negative, with value 23 − 24 = −1.
2 — Find the mistake
(a) The mistake is on Line 3 (carried into Line 4).
(b) "ADD 6" is wrong. At n = 1 the formula must equal the first term: 4(1) + c = 6, so c = 2, not 6. The student confused "add to make 6" with "add 6".
(c) Corrected working: differences are all 4 → starts with 4n. For n = 1, need 4 + c = 6 → c = 2. Rule: term = 4n + 2.
Check n = 1: 4 + 2 = 6 ✓. n = 2: 8 + 2 = 10 ✓. n = 5: 20 + 2 = 22 ✓.
3 — Design your own sequence (sample solution)
Constant-difference solution: "add 5" each time. Sequence: 10, 15, 20, 25, 30. Position-to-term rule: term = 5n + 5. Check n = 1: 10 ✓; n = 5: 30 ✓.
A different solution (varying step): e.g. 10, 14, 18, 24, 30 (steps +4, +4, +6, +6). First term 10 ✓; 5th term 30 ✓. The term-to-term rule isn't constant, so there's no single linear position-to-term rule — but the two end conditions are still met.
Other valid examples: 10, 12, 16, 22, 30 (steps +2, +4, +6, +8 — square-style growth); 10, 20, 15, 25, 30; or any sequence whose 1st term is 10 and whose 5th term is 30.
Marking: 1 mark for stating a valid term-to-term rule; 1 mark for correct first five terms; 1 mark for a verified position-to-term rule; 1 mark for the bonus second sequence with different structure.