A recurrence relation defines each term of a sequence from the one before, uₙ₊₁ = f(uₙ), starting from a given first term. Some recurrences repeat in a fixed cycle; these are periodic sequences, and the length of the cycle is the order. Sums of periodic sequences are found by grouping whole cycles, and linear recurrences of the form uₙ₊₁ = puₙ + q model processes such as savings or medicine doses.
Recurrence relations
- uₙ₊₁ = f(uₙ), with u₁ given
- Periodic of order k: uₙ₊ₖ = uₙ for all n
- Σ of a periodic sequence = (number of complete cycles) × (sum of one cycle) + (leftover terms)
- Linear model: uₙ₊₁ = puₙ + q
Generating terms
Start from the given first term and apply the rule repeatedly: each output becomes the next input. Work one term at a time and write each down. The rule may be linear, such as uₙ₊₁ = 3uₙ − 1, or non-linear, such as uₙ₊₁ = 1/uₙ or (uₙ − 3)².
Periodic sequences and their order
If the terms repeat in a fixed cycle, so that uₙ₊ₖ = uₙ for every n, the sequence is periodic and k is its order. Find the order by generating terms until the first term reappears. The order is the length of the repeating block, not the number of terms you happened to list.
Summing a periodic sequence
A periodic sequence is neither arithmetic nor geometric, so the standard sum formulae do not apply. Instead, find the sum of one complete cycle, multiply by the number of complete cycles, and add any leftover terms at the end. Check whether the number of terms is an exact multiple of the order.
Sums with an arithmetic or geometric part
Some sums combine a periodic sequence with a term that grows, such as Σ(uᵣ + 2r). Split the sum into two parts and use the right method for each: cycle grouping for the periodic part, and the arithmetic or geometric series formula for the rest. Then add the results.
Recurrence models
A linear recurrence uₙ₊₁ = puₙ + q models many real processes: p scales the amount (interest or decay) and q adds a fixed amount each period, such as a regular deposit or a new dose. Generate the terms one at a time, or use given data to find p and q by forming simultaneous equations.
Worked examples
Example 1. A sequence has u₁ = 2 and uₙ₊₁ = 3uₙ − 1. Find u₂, u₃ and u₄.
- u₂ = 3(2) − 1 = 5.
- u₃ = 3(5) − 1 = 14.
- u₄ = 3(14) − 1 = 41.
Answer: u₂ = 5, u₃ = 14, u₄ = 41.
Marks: M1 for u₂, A1 for u₃ and u₄.
Example 2. A sequence has u₁ = 4 and uₙ₊₁ = 5 − uₙ. Show that it is periodic and state its order.
- u₂ = 5 − 4 = 1, u₃ = 5 − 1 = 4, u₄ = 1, …
- The block 4, 1 repeats.
Answer: Periodic with order 2.
Marks: M1 for u₃ = u₁, A1 for order 2.
Example 3. Find the sum of the first 100 terms of the sequence 4, 1, 4, 1, …
- The order is 2, so 100 terms make 50 complete cycles.
- Each cycle sums to 4 + 1 = 5.
Answer: 50 × 5 = 250.
Marks: M1 for 50 × 5, A1 for 250.
Example 4. A periodic sequence runs 3, 7, 3, 7, … Find Σ (r = 1 to 20) of (uᵣ + 2r).
- Periodic part: 20 terms are 10 cycles, each summing to 10, so Σuᵣ = 100.
- Arithmetic part: Σ 2r = 2 × (20 × 21)/2 = 420.
- Add the two parts.
Answer: 520.
Marks: M1 for 100, M1 for 420, A1 for 520.
Example 5. A tree is 1.5 m tall. Its height after each year is modelled by hₙ₊₁ = 1.2hₙ + 0.3, with h₁ = 1.5. Find h₄.
- h₂ = 1.2(1.5) + 0.3 = 2.1.
- h₃ = 1.2(2.1) + 0.3 = 2.82.
- h₄ = 1.2(2.82) + 0.3 = 3.684.
Answer: h₄ = 3.684 m.
Marks: M1 for h₂, A1 for 3.684.
Common mistakes
- Using the arithmetic or geometric sum formula on a periodic sequence. Group complete cycles instead.
- Miscounting the order. The order is the length of the repeating block.
- Forgetting leftover terms. If the number of terms is not a multiple of the order, add the extra terms.
- Skipping a step when generating terms. Each term comes from the one before; write every term down.
- Not splitting a combined sum. Handle the periodic part and the arithmetic or geometric part separately.
How recurrence relations are examined
Recurrence questions appear within sequences and series on WMA12 (Pure 2) for Edexcel International A-Level students, and in the second year of 9MA0. Questions commonly:
- ask for the next few terms of a recurrence
- ask you to show a sequence is periodic and state its order
- find a sigma sum of a periodic sequence, sometimes with an added term
- find an unknown constant from a condition, such as the sequence being periodic
Multi-part recurrence questions, each with its mark scheme, are in the sequences and series exam questions. For more practice, use the sequences and series worksheet or the sequences and series revision engine.
These sums often combine with arithmetic series. Increasing and decreasing sequences link to geometric sequences.
Frequently asked questions
What is a recurrence relation?
A rule that gives each term of a sequence from the previous term, such as uₙ₊₁ = 2uₙ + 1, together with a starting value.
What does it mean for a sequence to be periodic?
Its terms repeat in a fixed cycle. The number of terms in the cycle is its order.
How do you sum a periodic sequence?
Sum one cycle, multiply by the number of complete cycles, and add any leftover terms.