IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~9 min read

Applications of Sequences & Series

Once you know the formulas for arithmetic and geometric sequences, the next skill is recognising which one a real-world problem is asking for. IB exams love application questions — savings plans, populations, doses of medicine, bouncing balls, stacking patterns. The maths is the same as the previous two notes; the work is in reading the situation correctly. Get the identification right and the rest is plug-and-chug.

📘 What you need to know

If a quantity changes by a fixed amount each step (added or subtracted), the situation is arithmetic. Use u_n = u₁ + (n − 1)d and the two S_n forms.
If a quantity changes by a fixed factor each step (multiplied, or grows/shrinks by a fixed percentage), the situation is geometric. Use u_n = u₁ · r^{n − 1} and the two S_n forms.
Percentage growth: increase by p% means r = 1 + p/100. Percentage decay: decrease by p% means r = 1 − p/100.
Always identify u₁ carefully — sometimes “month 1” means after one month of growth, sometimes the starting amount. Read the wording.
For “after how many … ?” questions, you’ll usually be solving an inequality — leading to a quadratic in n (arithmetic case) or a logarithm (geometric case).
If the formulas don’t apply (the situation isn’t arithmetic or geometric), you can always fall back on listing terms by hand — but you lose the speed of u_n and S_n shortcuts.

How to spot which type a problem is

Before you write any formulas, look at how the quantity changes from one term to the next. The single question to ask is: is the change a fixed amount, or a fixed factor?

Identify the type — fixed amount or fixed factor?

Recognising the type from real-world clues

The wording of the problem usually gives you the type immediately. Train your eye on the verbs and units used.

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Arithmetic clues

The quantity increases or decreases by a fixed amount each step.

Simple interest — same amount of interest each year
Stacking objects in rows of equal height
Saving an extra fixed amount every month
Counting dots or seats in a regular pattern
Time-based wages with a flat raise each year

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Geometric clues

The quantity is multiplied or scales by a fixed factor each step. Watch for percentages.

Compound interest — interest grows on interest
Population, bacterial, or viral growth
Radioactive decay or drug elimination
Bouncing ball heights (each bounce is a fraction of the previous)
Discount/depreciation by a percentage each period

Quick percentage rule: “increases by p%” → multiply by (1 + p/100). “decreases by p%” → multiply by (1 − p/100). e.g. up 8% → factor 1.08; down 15% → factor 0.85.

If you see “each year/month/day, the quantity grows by X amount“, that’s arithmetic. If you see “each year/month/day, the quantity grows by X percent” (or “doubles”, “halves”, “scales by factor”), that’s geometric. The word “percent” is the strongest tell.

A general problem-solving recipe

🧭 Recipe — any application question

Identify the type — fixed amount = arithmetic; fixed factor / percentage = geometric.
Identify u₁ — the very first term in the situation. Read the wording carefully (is “year 1” the start, or after one year of growth?).
Identify d or r — the constant difference, or constant ratio.
Decide what’s being asked: a specific term (use u_n), a running total (use S_n), or “how many steps until…” (set up an inequality and solve).
Write the answer in context — include units, and round only at the very end. “After 14 weeks” not just “14”.

If you write down “u₁ = …”, “d (or r) = …”, and “I’m finding u_n / S_n” before doing any algebra, you’ll catch most setup mistakes before they happen.

Worked examples

WE 1

Arithmetic — total saved

Sarah is saving for a holiday. In the first week she saves $50, and each week after that she saves $8 more than the previous week. How much has she saved in total after 12 weeks?

Step 1: Identify the type “$8 more each week” — fixed amount → arithmetic u₁ = 50, d = 8, n = 12 Step 2: We want a total → use S_n S₁₂ = (12/2)(2(50) + 11(8)) = 6 × (100 + 88) = 6 × 188 S₁₂ = $1128

WE 2

Arithmetic — when does the total exceed a threshold?

A theatre has 18 seats in the front row. Each row behind has 3 more seats than the row in front. After how many rows does the total number of seats first exceed 500?

Step 1: Identify and set up “3 more each row” → arithmetic u₁ = 18, d = 3 Step 2: Solve S_n > 500 (n/2)(2(18) + 3(n − 1)) > 500 (n/2)(33 + 3n) > 500 n(33 + 3n) > 1000 3n² + 33n − 1000 > 0 Step 3: Solve via GDC or quadratic formula n ≈ 13.57 (positive root) Step 4: Smallest integer above 13.57 check: S₁₃ = 468 < 500, S₁₄ = 525 > 500 ✓ After 14 rows always verify the boundary case — S₁₃ vs S₁₄ — to confirm

WE 3

Geometric — percentage growth

A startup launches with 600 users in month 1. Each subsequent month, the user base grows by 8% of the previous month. (a) Find the number of users in month 12. (b) During which month does the user base first exceed 2000?

Step 1: Identify the type “8% of previous” → geometric u₁ = 600, r = 1.08 (a) Find u₁₂ u₁₂ = 600 × 1.08¹¹ ≈ 600 × 2.332 = 1399.2 ≈ 1399 users in month 12 (b) Solve u_n > 2000 600 × 1.08^{n − 1} > 2000 1.08^{n − 1} > 10/3 (n − 1) ln(1.08) > ln(10/3) n − 1 > ln(10/3) / ln(1.08) ≈ 15.65 n > 16.65 Month 17 smallest integer above 16.65 — verify with month 16 (1903) vs month 17 (2056)

WE 4

Geometric — drug metabolism

A patient is given a 200 mg dose of medication. Each hour, 25% of the medication present in the body is metabolised. Find the amount remaining after 6 hours.

Step 1: Identify the type “25% lost each hour” → 75% remains → geometric u₁ = 200 (at hour 0), r = 0.75 Step 2: Amount after 6 hours = 200 × 0.75⁶ = 200 × 0.17798… = 35.596… ≈ 35.6 mg remaining “after n hours” usually corresponds to multiplying by r exactly n times — count carefully

WE 5

Sum to infinity — bouncing ball

A ball is dropped from a height of 5 m. After each bounce, it reaches 70% of the height of the previous bounce. Assuming it bounces infinitely many times, find the total vertical distance the ball travels.

Step 1: Visualise the motion drops 5, bounces up to 3.5, falls 3.5, bounces up to 2.45, falls 2.45, … Step 2: Total = initial drop + 2 × (sum of all bounce heights) bounce heights: 3.5, 2.45, 1.715, … geometric with u₁ = 3.5, r = 0.7 Step 3: Check |r| < 1 → converges → use S_∞ sum of bounces = 3.5 / (1 − 0.7) = 3.5 / 0.3 ≈ 11.67 m Step 4: Total distance = 5 + 2 × 11.67 = 5 + 23.33 ≈ 28.3 m the factor of 2 is because each bounce contributes “up + down” — easy to forget

💡 Top tips

Identify the type before writing any formula. “Fixed amount” = arithmetic; “fixed factor / percent” = geometric.
Be careful with “month 1” vs “after 1 month”. Sometimes u₁ is the starting amount; sometimes it’s the value after one step. Read closely.
For percentage problems: up p% → ratio (1 + p/100); down p% → ratio (1 − p/100). Don’t use r = 0.08 when 8% growth is meant — use 1.08.
For “after how many…?” questions, write the inequality, solve as a quadratic (arithmetic) or use logs (geometric), then check the integer-boundary case.
Keep units throughout. The final answer should match the units in the question — months, dollars, mg, metres.
For sum-to-infinity questions, always state “|r| < 1, so the series converges” — that’s a method mark.
If you can’t tell whether it’s arithmetic or geometric, compute the first 3 or 4 terms by hand and look at the pattern. Differences vs ratios will tell you immediately.

⚠ Common mistakes

Confusing simple and compound interest. Simple = same fixed amount each year (arithmetic); compound = same percentage each year (geometric). They give very different totals.
Using r = 0.08 instead of r = 1.08 for “8% growth”. The 0.08 is the change; 1.08 is the new total as a multiple of the old.
Off-by-one with u₁. If “year 1” already means after one year of growth, then u₁ isn’t the starting value — it’s the value after one step.
Forgetting the factor of 2 in the bouncing-ball type. Each bounce contributes both an up-trip and a down-trip.
Not checking convergence before applying S_∞. If |r| ≥ 1, no finite sum exists — the formula doesn’t apply.
Rounding mid-calculation. For Paper 2, keep full precision until the end; round only the final answer.
Forgetting to put the answer in context. If the question asks “in which month?”, “n = 17″ isn’t a complete answer — say “month 17”.

Application questions are where the IB tests whether you understand sequences as models of real-world change, not just as algebraic objects. Spend time on the identification step — it’s worth more than the algebra that follows.

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