SL1.3 Geometric Sequences and Series

SL 1.3 — Geometric Sequences, Series & Sigma Notation

Topic Description
Geometric Sequences Sequences where each term is multiplied by a constant ratio.
nth Term Formula Formula that allows calculation of any term without listing previous ones.
Geometric Series Sum of the first n terms of a geometric sequence.
Sigma Notation Notation for expressing repeated multiplication-based sums efficiently.
Applications Radioactive decay, population growth, investment growth, spread of epidemics.

1. Understanding Geometric Sequences

A geometric sequence is one where each term is obtained by multiplying the previous term by a constant ratio r.
If the first term is a, the sequence looks like:

a, ar, ar², ar³, …

Unlike arithmetic sequences (steady addition), geometric sequences model multiplicative growth or decay.
This makes them strongly connected to natural sciences, investment calculations, and spread-based phenomena.

🌍 Real-World Example:
Radioactive decay, bacterial reproduction, interest compounding in bank accounts, and viral spread all follow
geometric patterns because each stage depends on a percentage of the previous one.

🔍 TOK Perspective:
Geometric growth can feel unintuitive. How do we reconcile mathematical predictions (e.g., exponential spread)
with human intuition, which usually expects linear changes?

2. The nth Term of a Geometric Sequence

The general formula for the nth term is:

uₙ = a r⁽ⁿ⁻¹⁾

This formula is crucial for predicting future values — especially in finance or population modelling.

Example:

Sequence: 3, 6, 12, 24, …
a = 3, r = 2
The 10th term = 3 × 2⁹ = 1536.

🧠 Examiner Tip:
When a question gives two terms like u₃ and u₇, divide the equations to eliminate a.
This directly gives the common ratio r.

🔬 Science Connection (Physics):
Many nuclear decay processes follow the relation Nₙ = N₀ rⁿ.
This is identical in structure to geometric sequences.

3. Geometric Series — Sum of First n Terms

A geometric series is the sum of the first n terms of a geometric sequence.
The formula is:

Sₙ = a (1 − rⁿ) / (1 − r), for r ≠ 1

If |r| < 1, the terms get smaller and the series converges.
If |r| > 1, the terms grow rapidly and the series increases without bound.

Example:

Find S₅ for 4, 2, 1, 0.5, 0.25, …
a = 4, r = 1/2
S₅ = 4(1 − (1/2)⁵) / (1 − 1/2) = 7.75.

📝 Paper 2 Strategy:
Geometric sums frequently appear in modelling questions (epidemics, population growth).
Always check whether the question requires:

  • the sum of the series (Sₙ), or
  • a prediction using uₙ.

4. Sigma Notation for Geometric Sums

Sigma notation lets us express geometric sums compactly:

Σ ( a r⁽ⁿ⁻¹⁾ ) from n = 1 to k

To evaluate such expressions, identify:

  • first term a
  • ratio r
  • number of terms k

⚗️ IA Tip:
If your IA studies anything that grows or decays (savings, infections, chemical reactions),
geometric models outperform arithmetic ones.
Justify your choice mathematically — this strengthens Criterion E (Reflection).

5. Real-Life Applications

Geometric sequences appear when change happens by multiplying rather than adding:

  • compound interest
  • epidemic spread (R₀ modelling)
  • radioactive decay
  • charging of capacitors
  • population growth

❤️ CAS Idea:
Collect school attendance or social media engagement over time.
Determine whether behaviour follows a geometric or arithmetic pattern,
and present the findings visually.

🔍 TOK Discussion:
A finite geometric area (like the classical paradox of the infinite perimeter enclosing a finite area)
raises questions:
Does mathematics reflect reality, or does reality conform to mathematical abstraction?