Beyond Fibonacci: what ratio do other recurrence sequences approach?

AA HLAA SL Solid Also in: Art & music

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?

Adapt it: change the place, the data or the comparison until the question is yours.

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)

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

Investigate three-term recurrences and the “tribonacci” constant.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).