Unit 3 Topic 4 · week 8 · formula book uses t₁ (not a) · NO sum-of-series in this syllabus, and simultaneous questions are solved by setting equal, never elimination
The concept in one sentence: arithmetic sequences ADD the same amount each step, geometric sequences MULTIPLY by the same amount each step, and every question is just spotting which one you're in and using its rule.
The thing that makes it click: the first question is always "+ or ×?", asked out loud, before anything else. And for the rule: n − 1 counts the JUMPS, not the terms: getting to term 20 takes 19 jumps.
1 · Prerequisite check (30 seconds)
"What's 7 + 4 × 3?" (order of operations), and "What's 800 × 0.5, then × 0.5 again?"
Also connect backwards, not just check: he already OWNS geometric sequences without knowing it, compound interest was ×1.008 every step. Open with that: "you've been doing geometric sequences since week 1."
2 · The teaching path
Concrete: two training plans. Plan A: run 7 km, add 4 km each week. Plan B: 7 km, multiply by 1.5 each week. Write out five terms of each side by side, watch B explode. The + vs × distinction and the "geometric eventually wins" fact both land in one picture
The identification habit: a list of sequences, he sorts them saying "+4", "×2", "−5", "×0.85" out loud. Finding d or r by subtracting/dividing NEIGHBOURS becomes automatic here
The jump rule: to reach term 20 from term 1 you make 19 jumps. That's the whole n − 1:
Both costumes for every sequence: the rule (jump straight to term 20) and the recurrence (hop one step at a time). QCAA asks for both, often in the same question
Decay: the medicine dose (800 mg halving hourly). Multiplying by 0.5 IS geometric, decay is just r under 1, and depreciation from Unit 4 was this all along
The "set them equal" move: two plans cost the same when? 200 + 50n = 125 + 75n → n = 3. Equate, solve. This is the ONLY simultaneous-equation move the external ever needs (checked against four years of papers: elimination appears zero times)
Plus or times? Ask it before you touch anything.
Term 20 is 19 jumps away.
3 · Misconception catalogue
Looks like
Why brains do it
The fix script
Uses n instead of n − 1 (t₂₀ = 7 + 20 × 4)
20 terms feels like 20 additions
The jump picture: draw term 1 and term 3, count the arrows between them (2, not 3). "Jumps, not terms"
Writes a instead of t₁ (textbook letters)
Most textbooks and YouTube use a
"The formula book says t₁. Same creature, exam costume." Train with the book open, it's the letters he'll see under pressure
Treats 15% decay as subtract 15 each time
Percent OF feels like a fixed amount
"15% of a SHRINKING number shrinks too. Percent-of-changing = multiply. Fixed amount = add." Then connect: it's the ute from week 1, × 0.85
Finds r by subtracting neighbours
Subtracting is the habit from arithmetic
"+ or × first! If it's ×, DIVIDE neighbours to find r." 6 ÷ 4 = 1.5, confirm with the next pair
Reaches for elimination on a two-plan question
School taught elimination in Unit 2
"Both plans are already 'cost = ...'. Set them equal and solve. One line." Elimination never appears on this external
Indexing slip on decay ("after 4 hours" vs t₄)
t₁ is the hour-ZERO reading
"t₁ is the starting photo, before any time passes. After 4 hours you're at t₅." Label terms with their hour whenever time is involved. And name the clash out loud: sequences count from t₁ but week 1's finance recurrence counted from A₀ (check which convention the school's paper uses when it appears). The same ritual saves both: label every term with its time
4 · Questioning ladder
Rung
Ask
Listen for
Recall
"5, 9, 13... plus or times? What's d?"
"+4", instantly, and "×2" for 3, 6, 12
Do
"7 km week 1, add 4 each week. Week 20?"
7 + 19 × 4 = 83, with the 19 explained if asked
Explain back
"Why 19 and not 20?"
The jumps picture, in his own words
Transfer
"Your phone loses 30% of its value each year. What kind of sequence, and what's the multiplier?"
Geometric, × 0.7, ideally linking it to depreciation unprompted
5 · How QCAA examines it
Standard shape: a context (training, medicine, wages, value), then: write the rule, find a far term, write the recurrence, sometimes "which week does it first exceed...?" (Ans-key iteration answers that)
Scope cuts that save you time: NO sum of series (Sₙ was removed from these pages deliberately, it is not in the syllabus), NO elimination. Don't let a textbook drag either back in
The sequences-finance bridge is examiner-beloved: compound interest as geometric, straight-line depreciation as arithmetic, reducing-balance as geometric. Cross-labelling earns understanding marks
Formula book letters: t₁ and tₙ. The recurrence for finance uses Aₙ but the logic is identical, point at the rhyme
6 · Stuck scripts
Plus or times?
How many jumps from term 1?
Divide the neighbours. What do you get both times?
Which one is this: fixed amount, or percent of a changing thing?
7 · ADHD delivery notes
The "+ or ×" opener is another checklist-instead-of-open-question accommodation, same trick as the three-word chant
Lots of little identification reps beat two big problems: sorting sequences is fast, winnable and exactly what the drills serve
Lean hard on the week 1 connection: this week should FEEL like revision wearing a new shirt, say that out loud
The side-by-side training plans table is worth him building himself, the explosion of the geometric column is memorable