IA idea · Pure maths, number & proof
Beyond Fibonacci: what ratio do other recurrence sequences approach?
Research question
For sequences defined by uₙ₊₂ = a·uₙ₊₁ + b·uₙ, what limit does uₙ₊₁/uₙ approach, when does it exist, and can the limit be proved using the characteristic equation?
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Why it makes a good exploration
Everyone knows Fibonacci ratios approach the golden ratio. Generalising to any linear recurrence — and discovering when the ratio converges, oscillates or diverges — turns a cliché into a genuine investigation.
The mathematics you'll need
- Sequences and recurrence relations
- Quadratic characteristic equations
- Closed forms (Binet-style) proved by induction (HL)
- Limits and dominant roots
- Complex roots and oscillation (HL)
Where the data comes from
No data needed; use a spreadsheet to explore many (a, b) pairs; check known sequences on OEIS.
- OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Explore numerically for many (a, b).
- Conjecture the limit in terms of the characteristic roots.
- Derive a closed form and prove it by induction.
- Classify cases: real distinct, repeated, complex roots.
- Reflect on the golden ratio's special status.
Pitfalls that cost marks
- Only numerical tables.
- Ignoring cases where the limit does not exist.
- Copying Binet's formula without derivation.
Showing personal engagement
- Find your own recurrence with an interesting limit.
- Map the (a, b) plane into regions by behaviour.
- Explain where the golden-ratio myths go wrong.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Sequences and recurrence relations; Quadratic characteristic equations |
| AA HL | Good fit | Sequences and recurrence relations; Quadratic characteristic equations |
| AI SL | Not a natural fit | The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach. |
| AI HL | Not a natural fit | The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level. |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: only numerical tables; and ignoring cases where the limit does not exist. Say how each one affects your answer.
Use of mathematics (E)
SL: sequences and recurrence relations; quadratic characteristic equations; limits and dominant roots, each explained in your own words and carried out correctly.
HL adds: closed forms (Binet-style) proved by induction; complex roots and oscillation, each explained in your own words and carried out correctly.
How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Investigate three-term recurrences and the “tribonacci” constant.
Extending it for HL
This idea already has HL mathematics in it: Closed forms (Binet-style) proved by induction; Complex roots and oscillation. Make that the centre of the exploration and use it with full rigour.
See a complete IA, marked
Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete pure maths exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Sequences and recurrence relations, quadratic characteristic equations, closed forms (Binet-style) proved by induction (HL), limits and dominant roots and complex roots and oscillation (HL).
Where can I get data for this IA?
No data needed; use a spreadsheet to explore many (a, b) pairs; check known sequences on OEIS. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences) and Desmos graphing calculator.
Turn this idea into your IA
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