Recurrences for Maths Olympiad: Learning Path

1. R01

   Recursive sequences

   Compute terms and explain why initial conditions matter.

   Inside: Recursive sequences and initial conditions · Recursive versus explicit descriptions

2. R02

   Classifying recurrence relations

   Classify a recurrence to select an appropriate method.

   Inside: Order and linearity · Homogeneous and non-homogeneous recurrences

3. R03

   First-order linear recurrences

   Transform a first-order recurrence and verify its explicit formula.

   Inside: Arithmetic and geometric cases · Shifting a fixed point · Telescoping and variable forcing

4. R04

   Nonlinear recurrences and substitutions

   Find a substitution suggested by the recurrence rather than guessing at random.

   Inside: Reciprocal substitutions · Factoring a recurrence · Invariants and domain restrictions

5. R05

   Second-order linear recurrences

   Solve a second-order recurrence and justify the resulting expression.

   Inside: Characteristic roots · Distinct and repeated roots · Initial-value verification

6. R06

   Higher-order linear recurrences

   Explain how the order and root multiplicities control the general form.

   Inside: Higher-order recurrences · Repeated roots and enough initial conditions

7. R07

   Non-homogeneous recurrences

   Choose a valid particular form and adjust it when it overlaps the homogeneous part.

   Inside: Particular plus homogeneous solutions · Resonance · Verifying the combined solution

These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.