IB Maths AI SLTopic 1 — Sequences & SeriesPaper 1 & 2Real-world~7 min read
Applications of Sequences & Series
First decide whether the situation is arithmetic (adds/subtracts the same amount) or geometric (multiplies/divides by the same factor). Then pick the right formula and plug in. That’s the whole skill.
📘 How to spot which is which
Arithmetic — same amount added or subtracted each step. Words like “increases by $X”, “gets Y more each year”, “X fewer each row”.
Geometric — multiplied or divided by the same factor each step. Words like “X% per year”, “doubles”, “halves”, “grows at rate r”.
Percentages → multipliers: +5% means × 1.05; −20% means × 0.8; “loses 8%” means × 0.92.
Half-life = the time for the quantity to halve, so r = 0.5 per half-life period.
“Total accumulated” → use Sn. “How much in year/period n” → use un.
“When does it reach/exceed X?” → solve for n. Often needs logs (geometric) or a quadratic (arithmetic sum).
Arithmetic or geometric? — decide first
Identify the type FIRST, then pick the right formula. Mixing them up is the most common cause of lost marks on these questions.
🧭 Recipe — any application problem
Identify the type: fixed amount → arithmetic; fixed factor or percentage → geometric.
Find u1 and d or r from the question.
Decide what’s asked: a specific term (un), a total (Sn), or a time/position (n).
Plug into the right formula; use the GDC for tricky arithmetic.
Round sensibly: money 2 d.p., people / years whole numbers, otherwise 3 s.f.
Worked examples
WE 1
Arithmetic — salary raises
Maya starts a job at $42 000 per year and receives a $2 500 raise each year. (a) Find her salary in year 10. (b) Find her total earnings over the first 10 years.
A display stacks cans in rows. The bottom row has 18 cans, and each row above has 2 fewer cans. The stack has 9 rows. How many cans are used in total?
Identify (arithmetic, decreasing — fixed loss per row)u_1 = 18, d = −2, n = 9Use S_nS_9 = 9/2 × (2(18) + 8(−2)) = 9/2 × (36 − 16) = 9/2 × 20S9 = 90 canscheck the top row: u_9 = 18 + 8(−2) = 2 cans ✓
WE 3
Geometric — compound interest
$8 000 is deposited in a bank account that pays 6% compound interest per year. Find the balance after 12 years, to 2 d.p.
Identify (geometric — fixed % each year)multiplier = 1 + 6% = 1.06After 12 years (12 multiplications)balance = 8000 × 1.06^12 = 8000 × 2.0122…balance = $16 097.57+6% means × 1.06, never × 0.06.
WE 4
Geometric decay — radioactive
A 500 g sample of a radioactive substance decays at 8% per year. Find its mass after 15 years, to 3 s.f.
Identify (geometric, decreasing — fixed % loss)loses 8% = keeps 92% each yearu_1 = 500, r = 0.92After 15 yearsmass = 500 × 0.92^15 = 500 × 0.2863… = 143.15…mass ≈ 143 g (3 s.f.)“loses 8%” → multiplier is 0.92 (what’s left), not 0.08.
WE 5
Geometric — half-life
A drug dose of 240 mg has a half-life of 8 hours in the body. How much remains after 32 hours?
Identify (geometric — halves every period)multiplier per 8 hours: r = 0.5Count half-lives in 32 hours32 ÷ 8 = 4 half-livesApplyamount = 240 × 0.5^4 = 240 × 0.0625amount = 15 mgmatch the multiplier to the period — r = 0.5 per HALF-LIFE, not per hour.
WE 6
Find when a population exceeds a target (uses logs)
A town has 12 000 people and grows at 3% per year. After how many whole years will the population first exceed 20 000?
Set up (geometric, r = 1.03)12 000 × 1.03^n > 20 0001.03^n > 20 000 / 12 000 = 5/3Solve using logsn > log(5/3) / log(1.03)n > 17.28…Smallest whole nat n = 17: 12 000 × 1.03^17 = 19 834 (still ≤ 20 000)at n = 18: 12 000 × 1.03^18 = 20 429 (exceeds)n = 18 yearsalways check the integer either side — the question asks for whole years.
💡 Top tips
Read for the keyword: “fixed amount” / “the same each” → arithmetic. “Per cent” / “rate” / “factor” / “doubles” / “halves” → geometric.
Simple vs compound interest: simple interest is arithmetic (fixed $ each year); compound is geometric (×(1 + rate)).
Percent change → multiplier: +x% means × (1 + x/100); −x% means × (1 − x/100).
Match index to the question: “after 5 years” usually means 5 multiplications (geometric) or after the 5th term (arithmetic). Sketch the first 2 terms to check.
⚠ Common mistakes
Using arithmetic for a percentage: “grows by 5% each year” is GEOMETRIC, not arithmetic.
Using the percent itself as r: 8% loss → r = 0.92, NOT 0.08.
Off-by-one in the exponent: “after 5 years” with compound interest typically means × r5 from year 0 — not r4.
Forgetting to round properly: money to 2 d.p., people / years to a whole number, otherwise 3 s.f.
That wraps Sequences & Series. Up next in AI SL Topic 1: Financial Mathematics — compound interest done properly, with all the formula-booklet variations, plus annuities and amortisation.
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