IB Maths AI SLTopic 1 — Sequences & SeriesPaper 1 & 2In formula booklet~7 min read
Geometric Sequences & Series
A geometric sequence multiplies by the same number each step. That fixed multiplier is the common ratio, r. Same idea as arithmetic — but with × instead of +.
📘 What you need to know
Common ratio r: divide any term by the previous one (r = u2 / u1). Same value between every pair.
nth term formula: un = u1 × rn − 1.
Sum of first n terms — two versions, both in the formula booklet: Sn = u1(rn − 1) / (r − 1) — easier when r > 1. Sn = u1(1 − rn) / (1 − r) — easier when r < 1.
r > 1: increasing. 0 < r < 1: decreasing. r < 0: alternating sign.
Given two terms: divide them to eliminate u1 and solve for r.
Finding n usually needs logs (covered in the Logarithms note).
Finding the nth term
nth term — formula bookletun = u1 × rn − 1
Each step multiplies by r. To jump straight to u8 without listing all the doublings, raise r to the (n − 1)th power.
Finding the sum Sn
Sum of n terms — both in formula bookletSn = u1(rn − 1)r − 1 or Sn = u1(1 − rn)1 − r
Which version to use? Pick the one that keeps the numerator and denominator positive. For r > 1 the LEFT version (rn − 1) is cleaner. For 0 < r < 1 the RIGHT version (1 − rn) is cleaner. They give the same answer either way.
🧭 Recipe — solve any geometric problem
Identify u1 and r: divide any term by the previous one to get r.
To find a specific term: plug into un = u1 × rn − 1.
Given two terms (not consecutive): divide them — u5/u2 = r3 — then take the appropriate root.
To find a sum: plug into Sn; pick the version that suits your r.
To find n: rearrange to rn = value, then use logs.
Worked examples
WE 1
Find a specific term
A geometric sequence has first term 3 and common ratio 2. Find u8.
Find the sum of the first 10 terms of a geometric sequence with u1 = 80 and r = 0.75. Give your answer to 3 s.f.
r < 1, use S_n = u_1(1 − r^n)/(1 − r)S_10 = 80 × (1 − 0.75^10) / (1 − 0.75) = 80 × (1 − 0.0563…) / 0.25 = 80 × 0.9437… / 0.25 = 301.97…S10 ≈ 302 (3 s.f.)use the (1 − r^n) version when r is a fraction or decimal — keeps numbers positive.
WE 6
Real-world — depreciation
A car costs $30 000 new and loses 20% of its value each year. Find its value after 5 years.
Identify geometric sequenceloses 20% = keeps 80% of value each yearu_1 = $30 000 (start), r = 0.8After 5 years means 5 multiplicationsvalue = 30 000 × 0.8^5 = 30 000 × 0.32768value = $9830.40“loses 20%” → multiply by 0.8, NOT by 0.2. The 0.2 is what’s GONE; the 0.8 is what’s LEFT.
💡 Top tips
Spot geometric when quantities multiply by a fixed factor — compound interest, population growth, radioactive decay, depreciation.
Find r by dividing, never subtracting. r = u2/u1.
For percentage changes: +5% → r = 1.05; −20% → r = 0.8. Multiplier = 1 + (percent change as a decimal).
Pick the cleaner sum formula: r > 1 → (rn − 1); r < 1 → (1 − rn).
⚠ Common mistakes
Subtracting terms to find r: r is a RATIO — divide, don’t subtract.
Using rn instead of rn−1: u5 = u1 × r4, not r5. There are (n − 1) multiplications to reach the nth term.
Wrong multiplier for percent loss: “loses 20% each year” → r = 0.8, not r = 0.2.
Mixing arithmetic and geometric formulae: arithmetic uses d and +; geometric uses r and ×. Don’t confuse them.
Up next: Applications of Sequences & Series — putting arithmetic and geometric to work on real exam scenarios: compound interest, savings plans, depreciation, populations, and patterns.
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