Recurrence relations
Objective. Generate terms and reason about long-term behaviour.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Recurrence relations define each term using one or more previous terms, such as uₙ₊₁=f(uₙ), allowing sequences to be generated step by step from a starting value. Example: uₙ₊₁=3uₙ−2 with u₁=1 gives u₂=1, u₃=1, so the sequence is constant at the fixed point where uₙ₊₁=uₙ. Long-term behaviour depends on whether terms converge to a fixed point, oscillate, or diverge, found by iterating or solving L=f(L) for the limit L.
Worked example
A recurrence relation is uₙ₊₁=0.5uₙ+3 with u₁=2. Find u₂, u₃ and the long-term limit.
- 1Compute u₂=0.5(2)+3=4
- 2Compute u₃=0.5(4)+3=5
- 3Find the limit L by solving L=0.5L+3, giving 0.5L=3
- 4Solve for L: L=6
Answer: u₂=4, u₃=5, limit L=6
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Start practisingCommon mistake. Students assume a recurrence always converges and calculate a fixed point even when the sequence actually diverges; always iterate a few terms numerically to check the trend before trusting a fixed-point answer.
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