Geometric Sequences & Growth Models — SL 1.4 Geometric sequences and series; growth and decay models - IB Mathematics Applications and Interpretation

SECTION 1

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.

c. 300 BCE • Euclid's Elements

Euclid formally defined geometric progressions in Book IX of the Elements, proving key properties about summing terms in a constant-ratio sequence.

1494 • Pacioli & Compound Interest

Luca Pacioli published problems on compound interest in Summa de Arithmetica, showing that money growing at a fixed percentage rate forms a geometric sequence.

1798 • Malthus on Population

Thomas Malthus argued that populations grow geometrically while food supplies grow arithmetically, sparking worldwide debate on sustainability and resources.

1903 • Rutherford & Radioactive Decay

Ernest Rutherford described radioactive half-life using geometric decay models, showing that the amount of a radioactive substance halves in equal time intervals.

2020s • Viral Spread Models

During the COVID-19 pandemic, geometric growth models were widely used to project case counts, illustrating how quickly exponential processes can escalate.

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.

SECTION 2

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)

The starting value of the sequence. Every subsequent term is built from u₁ by repeated multiplication. In IB notation, u₁ is used instead of 'a'.

Common Ratio (r)

The factor you multiply by each time. Find it by dividing any term by its predecessor: r = u_n+1 ÷ u_n. If r > 1, the sequence grows; if 0 < r < 1, it decays.

General Term (uₙ)

Any term in the sequence can be found directly using the formula u_n = u₁ × rⁿ⁻¹, without calculating every term before it.

Geometric Series (Sₙ)

The sum of the first n terms. Useful when you need a cumulative total — for example, total savings after n deposits or total distance covered.

Growth & Decay Models

Real-world applications where a quantity changes by a fixed percentage each period: compound interest, depreciation, population change, and radioactive decay.

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KEY TAKEAWAY

Think of a geometric sequence like a photocopier with a zoom setting. If you set the copier to 120%, each copy is 1.2 times the previous one — the image grows rapidly. Set it to 80%, and each copy shrinks. The zoom percentage is your common ratio, and the original document is your first term. No matter how many copies you make, each one depends on that same fixed zoom factor.

SECTION 3

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 blue growth curve (r = 2) shows terms doubling each step, rising steeply by n = 6. The pink decay curve (r = 0.5) shows terms halving, approaching zero asymptotically. Both curves are mirror images of exponential behaviour.

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.

SECTION 4

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.

GENERAL (NTH) TERM

uₙ = u₁ × r ⁿ⁻¹

u_n = the nth term, u₁ = first term, r = common ratio, n = term number. Use this to find any single term directly.

SUM OF FIRST n TERMS (r ≠ 1)

Sₙ = u₁ × (rⁿ − 1) / (r − 1)

S_n = sum of the first n terms. An equivalent form is S_n = u₁ × (1 − rⁿ) / (1 − r). Use whichever avoids negatives in your calculation.

SUM TO INFINITY (|r| < 1)

S∞ = u₁ / (1 − r)

Only valid when |r| < 1 (the terms shrink toward zero). The infinite sum converges to a finite value. If |r| ≥ 1 the series diverges and this formula cannot be used.

COMPOUND GROWTH / DECAY MODEL

FV = PV × (1 + r/100)ⁿ

FV = future value, PV = present (initial) value, r = percentage change per period (positive for growth, negative for decay), n = number of periods. This connects geometric sequences to real-world models.

💡

IB Formula Booklet Tip

The general term and series sum formulas are provided in the IB formula booklet, so you don't need to memorise them. However, you do need to know when and how to apply each one. Practise identifying whether a problem asks for a single term (use u_n) or a total (use S_n).

SECTION 5

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

Common geometric growth and decay scenarios with their common ratios.

This classification chart shows five cases based on the value of r. The two convergent cases (|r| < 1) allow us to compute a finite sum to infinity. Growth and oscillating-growth cases diverge, meaning the partial sums increase without bound.

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.

SECTION 6

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.

Car Depreciation Problem

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Step 1 — Identify the Model

The car loses a fixed percentage each year, so this is a geometric decay problem. The initial value (u₁ or PV) is $24 000. The annual depreciation rate is 12%, so the common ratio is r = 1 − 0.12 = 0.88.
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Step 2 — Write the General Term

Using the compound model: FV = PV × (1 − 12/100)ⁿ = 24 000 × 0.88ⁿ. The value after n years is the (n + 1)th term of the geometric sequence, or equivalently FV = 24 000 × 0.88ⁿ where n = number of years.
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Step 3 — Calculate Value After 5 Years

FV = 24 000 × 0.88⁵. First calculate 0.88⁵: 0.88² = 0.7744, 0.88⁴ = 0.7744² = 0.59969536, 0.88⁵ = 0.59969536 × 0.88 ≈ 0.52773. Therefore FV ≈ 24 000 × 0.52773 ≈ 12 665.52.

Value after 5 years ≈ $12 665.52
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Step 4 — Calculate Total Loss

Total loss = Original price − Value after 5 years = 24 000 − 12 665.52 = 11 334.48.

Total depreciation over 5 years ≈ $11 334.48
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Step 5 — Interpret the Result

The car has lost roughly 47% of its value in five years. Notice that the total loss is not simply 5 × 12% = 60%, because depreciation compounds — each year's 12% loss is taken from a smaller value than the year before. This is the fundamental difference between simple and compound percentage change.

SECTION 7

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

Comparison of arithmetic and geometric sequences.

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KEY TAKEAWAY

If the amount being added each period stays the same (e.g., $50 per month), it's arithmetic. If the rate of change stays the same (e.g., 5% per year), it's geometric. On the IB exam, look for keywords like 'percent', 'doubles', 'halves', or 'by a factor of' — these are geometric signals. Words like 'each time $x more' or 'increases by a fixed amount' indicate arithmetic.

SECTION 8

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 = Ae^kt, 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 = N₀ $e^{−λt}$
Fixed percentage growth	Logistic growth models add a carrying capacity, making long-run predictions more realistic

How SL 1.4 concepts extend into higher-level mathematics.

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A sequence begins 5, 15, 45, 135, … Explain how you know this is a geometric sequence rather than an arithmetic one. State the values of u₁ and r.

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To test whether a sequence is geometric, check whether the ratio between consecutive terms is constant. Here 15 ÷ 5 = 3, 45 ÷ 15 = 3, and 135 ÷ 45 = 3. Because the ratio (not the difference) is constant, this is a geometric sequence. If it were arithmetic, the difference between consecutive terms would be constant — but 15 − 5 = 10 while 45 − 15 = 30, so it is not arithmetic. The first term is u₁ = 5 and the common ratio is r = 3.

Question 2: BASIC CALCULATION

Find the 8th term of the geometric sequence where u₁ = 6 and r = 2.

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Use the general term formula: u₈ = u₁ × r⁸⁻¹ = 6 × 2⁷. Calculate 2⁷ = 128, so u₈ = 6 × 128 = 768. The 8th term is 768. Notice how large the term becomes after only 8 steps — this illustrates the rapid acceleration of geometric growth when r > 1.

Question 3: INTERMEDIATE

The 3rd term of a geometric sequence is 36 and the 6th term is 972. Find the first term and the common ratio.

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From the general term formula: u₃ = u₁ × r² = 36 and u₆ = u₁ × r⁵ = 972. Divide u₆ by u₃ to eliminate u₁: (u₁ × r⁵) ÷ (u₁ × r²) = r³ = 972 ÷ 36 = 27. Therefore r³ = 27, giving r = 3. Substitute back into u₃ = u₁ × 9 = 36, so u₁ = 36 ÷ 9 = 4. The first term is 4 and the common ratio is 3. This technique of dividing two known terms to eliminate u₁ is a standard IB strategy.

Question 4: APPLIED

A city's population is 50 000 and it grows at 2.5% per year. (a) Write a model for the population after n years. (b) Find the population after 10 years (to the nearest whole number). (c) How many complete years until the population first exceeds 80 000?

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(a) This is geometric growth with PV = 50 000 and r = 1.025. The model is P(n) = 50 000 × 1.025ⁿ. (b) P(10) = 50 000 × 1.025¹⁰ ≈ 50 000 × 1.28008 ≈ 64 004. The population after 10 years is approximately 64 004. (c) We need 50 000 × 1.025ⁿ > 80 000, so 1.025ⁿ > 1.6. Taking logarithms: n > ln(1.6) ÷ ln(1.025) ≈ 0.4700 ÷ 0.02469 ≈ 19.03. Since n must be a whole number, the population first exceeds 80 000 after 20 complete years. Using logarithms to solve for the exponent is an essential skill for these problems.

Question 5: CRITICAL THINKING

An infinite geometric series has a first term of 12 and a sum to infinity of 48. Find the common ratio. Then explain why a geometric series with u₁ = 12 and r = 1.2 could never have a finite sum, no matter how many terms you include.

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Using S∞ = u₁ / (1 − r): 48 = 12 / (1 − r), so 1 − r = 12 / 48 = 0.25, giving r = 0.75. We can verify: |0.75| < 1, confirming convergence. For the second part, when r = 1.2, each term is 20% larger than the previous one. The terms grow without bound (u₂ = 14.4, u₃ = 17.28, etc.), so the partial sums keep increasing and never settle at a finite value. Mathematically, the S∞ formula requires |r| < 1 because only then do the terms approach zero, allowing the running total to stabilise. A series with |r| ≥ 1 diverges, meaning the sum grows toward infinity. This distinction between convergence and divergence is fundamental to understanding when geometric models give meaningful long-run predictions.

SUMMARY

Lesson Summary

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.