Historical Context & Motivation
Humans have recognized patterns of repeated multiplication for thousands of years. Ancient merchants noticed that doubling grain stores each harvest season produced quantities that grew astonishingly fast, while mathematicians in classical civilizations studied the properties of numbers that share a constant ratio between successive terms. The concept we now call a geometric sequence — a list of numbers where each term is obtained by multiplying the previous one by a fixed value — became one of the most powerful tools in mathematics for modelling real-world phenomena.
From compound interest calculations that bankers rely on daily to the half-life equations that nuclear physicists use, geometric sequences sit at the heart of exponential growth and decay. Understanding how they work gives you a lens to interpret phenomena that increase or decrease by a constant percentage over equal time intervals.
The central question this topic addresses is deceptively simple: what happens when a quantity is repeatedly multiplied by the same factor? The answer unlocks models for finance, biology, physics, and technology — and it all starts with understanding the common ratio.
Core Principles & Definitions
A geometric sequence is built on a single, elegant rule: multiply each term by a constant value, called the common ratio (r), to get the next term. This distinguishes it from an arithmetic sequence, where you add a constant difference. Whether r is greater than 1 (growth), between 0 and 1 (decay), or negative (alternating) determines the behaviour of the entire sequence.
First Term (u₁)
Common Ratio (r)
General Term (uₙ)
Geometric Series (Sₙ)
Growth & Decay Models
Visualising Geometric Growth & Decay
The diagram below plots two geometric sequences on the same axes. The blue curve shows geometric growth with u₁ = 2 and r = 2 (the terms double each step), while the pink curve shows geometric decay with u₁ = 64 and r = 0.5 (the terms halve each step). Notice how the growth curve accelerates upward while the decay curve approaches zero but never quite reaches it.
The key visual insight is the shape of each curve. Growth sequences produce a J-shaped curve — the increases seem small at first but become enormous. Decay sequences produce a curve that drops quickly then flattens out, approaching zero without ever reaching it. This flattening effect is why we say the values approach an asymptote (a line the curve gets infinitely close to but never touches). Both behaviours arise from the same formula — only the value of r differs.
Mathematical Framework
The IB syllabus for SL 1.4 centres on four key formulas. Each formula builds on the idea of repeated multiplication by the common ratio. Make sure you can identify which formula to use based on whether a problem asks for a single term, a sum, or a real-world percentage change.
Growth Models, Decay Models & Classification
In the real world, geometric sequences appear whenever a quantity changes by a fixed percentage in each equal time period. If a car loses 15% of its value every year, the value after each year forms a geometric sequence with r = 0.85. If a bacterial colony grows by 30% every hour, r = 1.30. The table below classifies the most common scenarios you will encounter on the IB exam.
| Scenario | Growth or Decay? | Common Ratio (r) | Example |
|---|---|---|---|
| Compound interest | Growth | 1 + (rate/100) | 5% annual interest → r = 1.05 |
| Population increase | Growth | 1 + (rate/100) | 3% annual growth → r = 1.03 |
| Depreciation | Decay | 1 − (rate/100) | 20% loss per year → r = 0.80 |
| Radioactive half-life | Decay | 0.5 (per half-life) | 50% decays each period → r = 0.5 |
| Drug concentration | Decay | 1 − (rate/100) | 40% eliminated per hour → r = 0.60 |
A quick way to remember: if the absolute value of r is less than 1, the terms get smaller and the series converges; if the absolute value of r is greater than or equal to 1, the terms stay the same size or grow, and the series diverges. On the IB exam, always state whether the series converges before attempting to use the S∞ formula — this is worth marks.
Worked Example — Compound Depreciation
A car is purchased for $24 000. It depreciates by 12% each year. Find (a) its value after 5 years, and (b) the total loss in value over those 5 years.
Geometric vs Arithmetic — Strengths & Limitations
One of the most common exam pitfalls is confusing arithmetic and geometric sequences. Both describe predictable patterns, but they model fundamentally different situations. The table below highlights the key differences, strengths, and limitations of each type.
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule | Add a constant (d) | Multiply by a constant (r) |
| General term | uₙ = u₁ + (n − 1)d | uₙ = u₁ × r ⁿ⁻¹ |
| Graph shape | Straight line (linear) | Exponential curve |
| Best for modelling | Constant additions — salaries, savings with fixed deposits | Percentage change — interest, depreciation, population |
| Sum to infinity | Never converges | Converges only when |r| < 1 |
| Limitation | Cannot model accelerating change | Assumes constant percentage — unrealistic over very long periods |
Connection to Advanced Models
The geometric sequence model you learn at SL 1.4 is actually the discrete (step-by-step) version of the continuous exponential function y = Aekt, which you may encounter at HL level or in science classes. The table below shows how key ideas at SL connect to more advanced theory.
| SL 1.4 Concept | Advanced Extension |
|---|---|
| Common ratio r | Becomes the base of the exponential function, eᵏ, where k is a continuous growth rate |
| Compound interest formula | Extends to continuous compounding: A = Pe^(rt), derived using limits |
| Sum to infinity S∞ | Connects to convergence of infinite series (Taylor series, power series) in HL calculus |
| Geometric decay (half-life) | Leads to differential equations: dN/dt = −λN, solved by N = N₀e^(−λt) |
| Fixed percentage growth | Logistic growth models add a carrying capacity, making long-run predictions more realistic |
You do not need to know these advanced forms for the SL exam, but being aware of them helps you understand why geometric sequences matter so much: they are the building blocks of exponential models that appear throughout science, finance, and engineering. Mastering SL 1.4 gives you a solid foundation to tackle these topics later.
Practice Problems
Lesson Summary
A geometric sequence is defined by a first term (u₁) and a common ratio (r), where each term equals the previous term multiplied by r. The general term formula uₙ = u₁ × r ⁿ⁻¹ lets you jump directly to any term, while the series sum formula Sₙ = u₁ × (rⁿ − 1)/(r − 1) gives the total of the first n terms. When |r| < 1, the series converges and the sum to infinity is S∞ = u₁/(1 − r).
In real-world applications, geometric sequences model compound growth (interest, population) when r > 1 and compound decay (depreciation, half-life) when 0 < r < 1, using the formula FV = PV × (1 + rate/100)ⁿ. Always identify whether you need a single term or a sum, check convergence before using S∞, and remember that the percentage change translates to r via r = 1 ± rate/100. These skills form the foundation of exponential modelling across mathematics and science.