Sequences and series form a compulsory Pure topic across Edexcel, AQA, OCR and OCR MEI A-Level Maths. The content splits into arithmetic sequences, geometric sequences, sigma notation, and recurrence relations, and while each has its own formulae, the real skill being tested is recognising which type of sequence a question is describing — because the formulae for arithmetic and geometric series look different and are not interchangeable.
This article works through each type in turn, with the formulae, a worked example, and the errors that recur most often, before finishing with sigma notation and recurrence relations.
Arithmetic sequences
An arithmetic sequence has a constant difference d between consecutive terms. If the first term is a, the nth term is \(a + (n-1)d\).
| Quantity | Formula |
|---|---|
| nth term | \(u_n = a + (n-1)d\) |
| Sum of first n terms | \(S_n = \dfrac{n}{2}\bigl(2a + (n-1)d\bigr)\) or \(S_n = \dfrac{n}{2}(a + l)\), where l is the last term |
Worked example
The first term of an arithmetic sequence is 5 and the tenth term is 41. Find the common difference and the sum of the first 20 terms.
- Use \(u_{10} = a + 9d\): \(41 = 5 + 9d\), so \(9d = 36\), \(d = 4\).
- Use \(S_n = \dfrac{n}{2}(2a + (n-1)d)\) with n = 20: \(S_{20} = \dfrac{20}{2}(2(5) + 19(4)) = 10(10 + 76) = 10 \times 86 = 860\).
Geometric sequences
A geometric sequence has a constant ratio r between consecutive terms. If the first term is a, the nth term is \(ar^{n-1}\).
| Quantity | Formula | Condition |
|---|---|---|
| nth term | \(u_n = ar^{n-1}\) | any r |
| Sum of first n terms | \(S_n = \dfrac{a(1-r^n)}{1-r}\) | \(r \neq 1\) |
| Sum to infinity | \(S_\infty = \dfrac{a}{1-r}\) | \(|r| < 1\) only |
Worked example
A geometric sequence has first term 8 and common ratio 0.5. Find the sum to infinity, and the smallest n for which the sum of the first n terms exceeds 15.9.
- Since \(|r| = 0.5 < 1\), the sum to infinity exists: \(S_\infty = \dfrac{8}{1-0.5} = 16\).
- Set up the inequality: \(\dfrac{8(1-0.5^n)}{1-0.5} > 15.9\), which simplifies to \(16(1-0.5^n) > 15.9\).
- Rearrange: \(1 - 0.5^n > 0.99375\), so \(0.5^n < 0.00625\).
- Take logs of both sides: \(n\ln(0.5) < \ln(0.00625)\). Since \(\ln(0.5)\) is negative, divide and flip the inequality: \(n > \dfrac{\ln(0.00625)}{\ln(0.5)} \approx 7.32\).
- So the smallest integer n is 8.
Identifying which type you have
- Check consecutive differences: if they are constant, it is arithmetic.
- Check consecutive ratios: if they are constant, it is geometric.
- A question describing 'growth by a percentage each year' or 'multiplied by' is geometric; one describing 'increases by a fixed amount' or 'added each time' is arithmetic.
- Context clues matter in modelling questions: compound interest and depreciation are geometric; a savings scheme adding the same fixed sum each month is arithmetic.
Sigma notation
Sigma notation, \(\sum_{r=1}^{n} u_r\), is shorthand for adding up terms of a sequence, and A-Level questions expect you to evaluate it directly and to write series in sigma notation yourself.
- The lower limit tells you where to start substituting r, and the upper limit where to stop, inclusive of both ends.
- \(\sum_{r=1}^{n} c = cn\) for a constant c — a common trap, since it is easy to assume this term contributes nothing.
- Sigma notation can wrap either an arithmetic or a geometric formula for u_r, so identify the type inside the sigma before applying a sum formula.
- Standard results, such as \(\sum_{r=1}^{n} r = \dfrac{n(n+1)}{2}\), are sometimes given in the formula booklet and sometimes expected to be derived — check your board's booklet.
Recurrence relations
A recurrence relation defines each term using the previous one (or two), such as \(u_{n+1} = 3u_n - 2\), with a given first term. These appear on every board and are tested in a few characteristic ways.
- Generating terms: substitute directly, term by term, being careful with the order of operations.
- Finding a limit (if it exists): as \(n \to \infty\), if the sequence converges to a limit L, then L must satisfy \(L = 3L - 2\) (replacing both \(u_{n+1}\) and \(u_n\) with L), which can be solved directly.
- Periodic sequences: some recurrence relations produce a sequence that repeats after a fixed number of terms; questions may ask you to find a specific term using the period rather than generating every term by hand.
| Step | Working |
|---|---|
| Relation | \(u_1 = 2\), \(u_{n+1} = \dfrac{1}{1-u_n}\) |
| u2 | \(\dfrac{1}{1-2} = -1\) |
| u3 | \(\dfrac{1}{1-(-1)} = \dfrac{1}{2}\) |
| u4 | \(\dfrac{1}{1-\tfrac{1}{2}} = 2\) |
| Conclusion | The sequence repeats with period 3, so \(u_{100} = u_{100 \bmod 3} = u_1 = 2\) (since 100 = 3×33+1) |
Common mistakes across the topic
- Using the arithmetic sum formula on a geometric sequence, or vice versa, because the sequence type was not checked first.
- Applying the sum to infinity formula without checking \(|r| < 1\).
- Forgetting to flip an inequality when dividing by a negative logarithm in 'find the smallest n' questions.
- Misreading sigma notation limits, particularly off-by-one errors when the lower limit is not 1.
- Assuming a recurrence relation converges without checking, especially in 'find the limit' questions.
A revision checklist
- Can you write down all four core formulae (arithmetic nth term and sum, geometric nth term and sum) from memory?
- Can you identify arithmetic versus geometric from a worded context, not just from a list of numbers?
- Can you solve a sum to infinity question, including stating the convergence condition explicitly?
- Can you handle an inequality involving \(r^n\), including the direction-flip when logs of a negative-log base are involved?
- Can you generate terms from a recurrence relation and find a limit, checking convergence first?