Historical Context & Motivation
Long before modern algebra existed, mathematicians noticed that some quantities grow (or shrink) by the same proportion at every step. Ancient Greek scholars studied these patterns in geometry and music, while later thinkers applied them to problems in commerce and astronomy. The idea of a geometric sequence — a list of numbers where each term is obtained by multiplying the previous term by a fixed value — has been a cornerstone of mathematics for over two millennia.
The central question this topic addresses is straightforward yet powerful: if every term in a sequence is a fixed multiple of the one before it, can we find any term directly and, even more usefully, the sum of many (or infinitely many) terms? The formulas you will learn in this lesson answer both questions elegantly.
Core Principles & Definitions
A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted r. The first term is denoted u1. For example, the sequence 3, 6, 12, 24, … has u1 = 3 and r = 2. Understanding these building blocks will unlock every formula in this lesson.
Common Ratio (r)
General (nth) Term
Finite Geometric Series (Sₙ)
Infinite Geometric Series (S∞)
Visual Explanation — Geometric Growth
Notice how the bars do not grow by the same amount — the jump from 2 to 4 is +2, while the jump from 16 to 32 is +16. However, the ratio between consecutive terms is always the same (each term is doubled). This multiplicative pattern, rather than an additive one, is what distinguishes a geometric sequence from an arithmetic one. When r > 1 the terms shoot upward; when 0 < r < 1 the terms decay toward zero; and when r is negative the terms alternate in sign.
Mathematical Framework
Three key formulas form the backbone of this topic in the IB syllabus. Each appears on the IB formula booklet, so your job is to understand when and how to use them. Below, u1 is the first term, r is the common ratio, and n is the number of terms.
Convergence vs. Divergence — A Closer Look
The behaviour of a geometric series depends entirely on the magnitude of the common ratio. When |r| < 1, each successive term becomes smaller and smaller, so the running total settles toward a fixed value — the series converges. When |r| ≥ 1, the terms either stay the same size or grow, so the sum increases without bound — the series diverges.
| Condition on r | Behaviour of Terms | Series Result |
|---|---|---|
| |r| < 1 (e.g., r = 0.5, −0.3) | Terms shrink toward 0 | Converges — use S∞ = u₁ / (1 − r) |
| |r| = 1 (r = 1 or r = −1) | Terms stay same size (or alternate sign) | Diverges — no finite sum |
| |r| > 1 (e.g., r = 2, −3) | Terms grow in magnitude | Diverges — no finite sum |
Worked Examples
Geometric vs. Arithmetic — Key Differences
Students often confuse geometric and arithmetic sequences because both involve a pattern between consecutive terms. The table below highlights the critical differences that will help you choose the correct approach on the IB exam.
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule between terms | Add a constant (common difference, d) | Multiply by a constant (common ratio, r) |
| General term | uₙ = u₁ + (n − 1)d | uₙ = u₁ × rⁿ⁻¹ |
| Sum formula | Sₙ = n/2 × (2u₁ + (n − 1)d) | Sₙ = u₁(rⁿ − 1)/(r − 1) |
| Growth pattern | Linear (constant increase/decrease) | Exponential (accelerating growth or decay) |
| Infinite sum? | Never converges (unless d = 0) | Converges only when |r| < 1 |
| Real-world example | Saving $50 each month | Compound interest at 5 % per year |
Connections to HL and Further Mathematics
The geometric series framework you have learned at SL level is the gateway to several powerful ideas explored in HL and university-level mathematics. Understanding where these extensions lead can deepen your appreciation of why geometric sequences matter so much.
| SL 1.3 Concept | HL / University Extension |
|---|---|
| Finite geometric sum Sₙ | Partial fraction decomposition and telescoping sums in HL calculus |
| Infinite sum S∞ when |r| < 1 | Power series, Taylor/Maclaurin series (HL Calculus), Fourier series (university) |
| Condition |r| < 1 for convergence | Ratio test and root test for general series convergence (university analysis) |
| Common ratio as a multiplier | Eigenvalues of matrices, geometric growth in differential equations |
One particularly elegant application is the derivation of the formula for compound interest. When you invest money at a fixed interest rate, your balance each year forms a geometric sequence. Summing regular deposits over time uses the finite geometric series formula directly. In HL, you will also encounter the binomial series, which generalises the idea of an infinite geometric sum to expressions like (1 + x)n for non-integer n, a tool with wide applications in physics and engineering.
Practice Problems
Lesson Summary
A geometric sequence is formed by multiplying each term by a constant called the common ratio (r). The general term formula uₙ = u₁ × rⁿ⁻¹ gives any term directly, while the finite sum Sₙ = u₁(rⁿ − 1)/(r − 1) adds the first n terms efficiently. When |r| < 1, the terms decay toward zero and the infinite sum S∞ = u₁/(1 − r) converges to a finite value.
On the IB exam, always identify u₁ and r first, choose the appropriate formula, and show substitution clearly. Remember: arithmetic means adding a constant, geometric means multiplying by a constant. These sequences appear in real-world contexts from compound interest to radioactive decay, making them one of the most widely applicable tools in the IB syllabus.