SL1.3 Geometric Sequences and Series

Topic Description
Geometric Sequences Sequences where each term is obtained by multiplying the previous term by a constant ratio.
nth Term Formula Formula used to calculate any term directly without listing earlier terms.
Geometric Series The sum of the first n terms of a geometric sequence.
Sigma Notation Compact notation used to represent repeated multiplication-based sums.
Applications Radioactive decay, population growth, compound interest, epidemic spread.

1. Understanding Geometric Sequences

A geometric sequence is a sequence where each term is formed by multiplying the previous term by a constant value called the common ratio r.

a, ar, ar2, ar3, ā€¦

  • Geometric sequences model multiplicative change, not additive change.
  • If |r| > 1, values increase rapidly (growth).
  • If 0 < |r| < 1, values decrease steadily (decay).

šŸŒ Real-World Connection

  • Compound interest grows by a fixed percentage each period, forming a geometric sequence.
  • Radioactive substances decay by losing a fixed fraction of atoms over equal time intervals.
  • Viral content spreads as each person shares with multiple others, creating exponential growth.

šŸ” TOK Perspective

  • Why does exponential growth often conflict with human intuition?
  • Do mathematical models describe reality, or only idealised versions of it?
  • At what point does relying on a model become misleading?

2. The nth Term of a Geometric Sequence

The nth term formula allows direct prediction of future values:

un = a rnāˆ’1

  • a is the first term.
  • r is the common ratio.
  • n is the term position.

Example:

3, 6, 12, 24 ā€¦ ā†’ a = 3, r = 2
u10 = 3 Ɨ 29 = 1536

šŸ§  Examiner Tip

  • Divide two given terms to eliminate a and solve directly for r.
  • Always check whether the ratio is greater than or less than 1.
  • Write the full substitution before evaluating.

šŸ”¬ Science Connection

  • Nuclear decay equations follow the same structure as geometric sequences.
  • Capacitor charging and discharging curves rely on exponential behaviour.
  • Population models in biology often assume geometric growth in early stages.

3. Geometric Series ā€” Sum of First n Terms

A geometric series adds the first n terms of a geometric sequence:

Sn = a(1 āˆ’ rn) / (1 āˆ’ r), r ā‰  1

  • If |r| < 1, the series converges.
  • If |r| > 1, the series grows rapidly.
  • The formula relies on cancellation of terms.

šŸ“ Paper 2 Strategy

  • Identify whether the question asks for un or Sn.
  • Clearly state values of a, r, and n.
  • Avoid substituting before simplifying powers.

šŸ“ IA Spotlight

  • Compare arithmetic and geometric models using real data.
  • Justify model choice mathematically and contextually.
  • Discuss limitations of exponential assumptions.