Sequences and series are one of the favourite topics of A-level examiners — and not by accident. It is a topic with few formulae, a lot of logic, and real applications in finance: savings, loans and compound interest. A student who knows five basic formulae and when to apply each one earns an almost guaranteed question. This guide takes you from the basic definition of a sequence, through arithmetic and geometric sequences, calculating sums, the sum to infinity, and on to the financial applications that appear in the exam — with no gaps and with full worked examples.
Sequences and series are the topic where you can practically 'guarantee' yourself a question in the A-level Pure paper. Unlike calculus, which demands command of dozens of rules, the topic rests on four basic formulae only — two for the arithmetic sequence, two for the geometric — plus one more for the sum to infinity. All the difficulty lies not in the formulae themselves but in two things: correctly identifying whether the sequence is arithmetic or geometric, and correctly translating a worded question into the formula. Sequences are taught in Year 12 in the first year of A level, right after algebra and functions are consolidated, and they serve as the basis for financial applications — compound interest, regular savings and loans — which are exactly the kind of questions examiners love. In this guide we go over everything you need: what a sequence is, the nth term and sum of an arithmetic sequence, the same for a geometric sequence, the sum to infinity, the financial applications, the mistakes that lose marks, and a focused practice plan.
What is a sequence — the basic terms
A sequence is an ordered list of numbers, each of which is called a 'term'. The first term is written a₁ (or u₁ in many A-level textbooks), the second a₂, and so on, and the general term in position n is written aₙ. The n is always a natural number (1, 2, 3, ...) and gives the position in the sequence, not the value. For example, in the sequence 5, 8, 11, 14 we have a₁ = 5, a₂ = 8, a₃ = 11 and a₄ = 14.
The two types of sequence studied at A level are the arithmetic sequence (in which the difference between consecutive terms is constant) and the geometric sequence (in which the ratio between consecutive terms is constant). Identifying the type is the first step in every question — and we will come back to it later, because that is where most students slip.
Arithmetic sequences — the nth term
In an arithmetic sequence you add the same fixed number each time, called the common difference and written d. If d is positive the sequence is increasing, if d is negative it is decreasing. Example: 3, 7, 11, 15, 19 — here a₁ = 3 and d = 4, because each term is 4 more than the one before.
The nth-term formula: aₙ = a₁ + (n − 1)·d. Why (n − 1) and not n? Because from the first term to the nth term you take n − 1 'steps' of d. To get from a₁ to a₅ you take 4 steps, not 5. This is the most common mistake in the topic, and we will return to it in the mistakes section.
Example: in an arithmetic sequence a₁ = 3 and d = 4. What is the tenth term? Substitute: a₁₀ = 3 + (10 − 1)·4 = 3 + 9·4 = 3 + 36 = 39. Notice that we multiplied by 9, not by 10.
Working backwards — finding d from two terms. If a₃ = 11 and a₇ = 23 are given, then between position 3 and position 7 there are 4 steps, so 4d = 23 − 11 = 12, giving d = 3. Now a₁ can be found: a₃ = a₁ + 2d → 11 = a₁ + 6 → a₁ = 5.
How to prove that a sequence is arithmetic
To prove that a sequence is arithmetic, show that the difference between any two consecutive terms is constant — that is, aₙ₊₁ − aₙ = constant, for all n. If the general expression for aₙ is given (for example as the result of a sum formula, or as an algebraic expression in n), the usual exam method is: (1) write aₙ₊₁ (substitute n + 1 for n in the expression), (2) calculate aₙ₊₁ − aₙ and simplify algebraically, (3) if the result is a fixed number that does not depend on n — the sequence is arithmetic, and the common difference d is exactly that number.
A full example: given aₙ = 3n + 2. Prove that the sequence is arithmetic and find d. Step 1: aₙ₊₁ = 3(n + 1) + 2 = 3n + 3 + 2 = 3n + 5. Step 2: aₙ₊₁ − aₙ = (3n + 5) − (3n + 2) = 3. Step 3: the result 3 is a constant that does not depend on n, so the sequence is arithmetic with d = 3.
Another example with a quadratic expression (to see when a sequence is not arithmetic): given aₙ = n². Calculate aₙ₊₁ − aₙ = (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1. The result 2n + 1 does depend on n (for example between a₁ and a₂ the difference is 3, and between a₂ and a₃ it is 5) — so the sequence is not arithmetic.
Sometimes the exam gives a sequence defined recursively (each term defined from the previous one), for example aₙ₊₁ = aₙ + 5. In that case the proof is immediate: aₙ₊₁ − aₙ = 5 straight from the definition, with no need to find a general expression — the sequence is arithmetic with d = 5 by definition.
The sum of an arithmetic series
Often the question asks not for a single term but for the sum of the first n terms, written Sₙ. The formula: Sₙ = n·(a₁ + aₙ) / 2. In words: the number of terms times the average of the first and last terms. This is the formula the young Gauss discovered when he added 1 to 100 in seconds: 100·(1 + 100)/2 = 50·101 = 5,050.
When you do not know aₙ but do know a₁ and d, you can substitute aₙ = a₁ + (n − 1)d to get a second sum formula: Sₙ = n·(2a₁ + (n − 1)·d) / 2. The two formulae are completely equivalent — choose according to what the question gives you. Both are on the A-level formula booklet, but you should know them anyway.
Example: sum the series 3 + 7 + 11 + ... + 39 (the same sequence as before, up to the tenth term). Here n = 10, a₁ = 3, a₁₀ = 39. Substitute: S₁₀ = 10·(3 + 39)/2 = 10·42/2 = 10·21 = 210.
| Quantity | Arithmetic sequence | Example (3, 7, 11, ...) |
|---|---|---|
| Rate of change | Constant difference d (adding) | d = 4 |
| nth term | aₙ = a₁ + (n − 1)·d | aₙ = 3 + (n − 1)·4 |
| Sum of n terms | Sₙ = n(a₁ + aₙ)/2 | S₁₀ = 210 |
| Increasing or decreasing | Increasing if d > 0, decreasing if d < 0 | Increasing (d = 4 > 0) |
Geometric sequences — the nth term
In a geometric sequence you multiply by the same fixed number each time, called the common ratio and written r. Example: 2, 6, 18, 54 — here a₁ = 2 and r = 3, because each term is 3 times the one before. Note the essential difference: in an arithmetic sequence you add, in a geometric one you multiply.
The nth-term formula: aₙ = a₁·r^(n−1). Again (n − 1) appears, by the same logic: from the first term to the nth term you multiply by r exactly n − 1 times. Example: in the sequence 2, 6, 18, ... what is the fifth term? a₅ = 2·3^(5−1) = 2·3⁴ = 2·81 = 162.
Finding r from two terms: divide. If a₂ = 6 and a₅ = 162, then a₅/a₂ = r³ (three steps) → 162/6 = 27 → r = 3. If the ratio satisfies 0 < r < 1 the sequence decays (each term smaller than the last), and if r > 1 it grows fast. If r is negative the terms alternate in sign.
The sum of a finite geometric series
The sum of the first n terms of a geometric sequence (when r ≠ 1): Sₙ = a₁·(rⁿ − 1) / (r − 1). There is also the equivalent form Sₙ = a₁·(1 − rⁿ) / (1 − r) — more convenient when r is less than 1, so that the numerator and denominator stay positive. Both forms give the same result.
Example: sum the first seven terms of 2, 6, 18, ... Here a₁ = 2, r = 3, n = 7. Substitute: S₇ = 2·(3⁷ − 1)/(3 − 1) = 2·(2,187 − 1)/2 = 2·2,186/2 = 2,186. (If r were equal to 1 the sequence would be constant and the sum would simply be n·a₁ — a rare case that needs checking separately.)
The sum to infinity of a geometric series
What happens if you add infinitely many terms of a geometric sequence? If |r| ≥ 1, the sum grows without limit and has no finite value. But if the ratio is less than 1 in absolute value, that is −1 < r < 1, the terms get smaller so fast that the sum converges to a finite number. This is the sum to infinity, and its beautiful formula is: S∞ = a₁ / (1 − r).
Example: 1 + 1/2 + 1/4 + 1/8 + ... Here a₁ = 1 and r = 1/2. Since |1/2| < 1 the series converges: S∞ = 1 / (1 − 1/2) = 1 / (1/2) = 2. Intuitively: you get closer and closer to 2 but never cross it — exactly the idea of Zeno's paradox.
A common reverse use in the exam: the sum to infinity is given and you are asked for r or a₁. For example, if an infinite geometric series converges to 12 and its a₁ is 8, then 12 = 8/(1 − r) → 1 − r = 8/12 = 2/3 → r = 1/3. Always confirm that the r you get is indeed in the interval (−1, 1).
Applications — compound interest and savings
Here sequences meet the real world, and examiners love it. Compound interest is the classic example of a geometric sequence: a sum invested that earns a fixed annual rate of interest grows every year by a constant factor. If you invest an amount P at an annual rate of r percent, after one year you have P·(1 + r/100), after two years P·(1 + r/100)², and in general after n years: P·(1 + r/100)ⁿ. This is a geometric sequence with common ratio 1 + r/100.
Example: £10,000 is deposited at an annual rate of 5%. How much is there after 3 years? The multiplier is 1.05, so the amount = 10,000 × 1.05³ = 10,000 × 1.157625 = £11,576.25. Notice that this is more than 10,000 + 3 × 500 = £11,500, which we would get with simple interest (arithmetic) — the difference is exactly the 'interest on the interest'.
A fixed monthly or annual saving leads to the sum of a geometric series. If a fixed amount is paid in at the start of every year and it earns interest, each payment grows according to how many years it has had to accumulate — and the total of all the payments is the sum of a finite geometric series, calculated with the Sₙ formula. In such questions the trick is to recognise that each payment is a term in a geometric sequence, and to add them with the sum formula.
When is it arithmetic? When the increase is a fixed amount (not a percentage). For example, a salary that rises by £300 every year is an arithmetic sequence; a salary that rises by 4% every year is a geometric sequence. The distinction 'fixed amount' versus 'percentage' is the key to identifying the type of sequence in a worded question.
Common mistakes that lose marks
Mistake 1 — confusing arithmetic with geometric. Students who notice that the sequence is 'growing' jump to a formula without checking whether the growth is by adding (arithmetic) or by multiplying (geometric). Always check: 2, 4, 6 is arithmetic (d = 2), but 2, 4, 8 is geometric (r = 2). Always run the identification test on three terms.
Mistake 2 — confusing n with (n − 1). This is the number-one error in the topic. In the nth-term formulae the index and the multiplier are (n − 1), not n. The 10th term of an arithmetic sequence is a₁ + 9d, not a₁ + 10d. In a geometric one it is a₁·r⁹, not a₁·r¹⁰. Always count steps, not terms.
Mistake 3 — using the sum to infinity without checking |r| < 1. If the ratio is greater than or equal to 1 in absolute value there is no finite sum, and substituting into the formula gives a wrong result (sometimes an absurd negative one). Always state the convergence condition before substituting.
Mistake 4 — confusing the sum Sₙ with the term aₙ. The question 'what is the tenth term' is completely different from 'what is the sum of the first ten terms'. Read the question twice and mark whether it asks for a single term or a sum. With sigma notation, check the lower limit too — a sum that starts at k = 0 has one more term than one starting at k = 1.
Mistake 5 — in compound interest, using the simple-interest formula. Compound interest is always geometric: multiply by (1 + r/100) each period. Adding P + n × (interest) belongs to simple interest only, which almost never appears at A level.
A focused practice plan
The sequences topic can be conquered thoroughly in about a fortnight of focused practice, around an hour a day. Here is the recommended order.
- Days 1-2: arithmetic sequences. Twenty exercises on finding the nth term and finding d from two terms, until the identification test and step counting become automatic.
- Days 3-4: the sum of an arithmetic series. Practise both sum formulae, including cases where you first need to find n or aₙ.
- Days 5-6: geometric sequences — nth term and ratio. Put the emphasis on indices and on cases with negative r or 0 < r < 1.
- Days 7-8: finite geometric sums. Practise both forms of the formula and choose the convenient one according to the size of r.
- Days 9-10: the sum to infinity. Every exercise starts with the convergence check |r| < 1, including reverse questions that return a₁ or r.
- Days 11-12: financial applications — compound interest and savings. Practise spotting 'fixed amount' versus 'percentage' in worded questions.
- Days 13-14: full past-paper questions under exam conditions (timed, no peeking at the solution), and self-marking against the mark scheme.
Summary
Sequences and series are among the most 'rewarding' topics at A level: few formulae, clear logic, and an almost guaranteed question. If you remember four formulae — nth term and sum for arithmetic (aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2), and the same two for geometric (aₙ = a₁·rⁿ⁻¹ and Sₙ = a₁(rⁿ − 1)/(r − 1)), plus the sum to infinity S∞ = a₁/(1 − r) — you are equipped for every question on the topic. Take care over the two things that trip students up: identifying the type of sequence correctly (adding versus multiplying), and counting steps correctly (n − 1, not n). Consistent practice on a variety of questions, and especially on the worded and financial ones, will make sequences your safe marks in the exam.
Frequently asked questions
How do you prove that a sequence is arithmetic?
Calculate aₙ₊₁ − aₙ (the difference between the next term and the current one) and simplify algebraically. If the result is a fixed number that does not depend on n — the sequence is arithmetic, and that number is the common difference d. For example, for aₙ = 3n + 2: aₙ₊₁ − aₙ = (3n + 5) − (3n + 2) = 3, a constant — so the sequence is proved arithmetic with d = 3.
How do you prove a sequence is arithmetic from a sum formula?
If the sum formula Sₙ is given, first find the general expression for aₙ (using aₙ = Sₙ − Sₙ₋₁ for n ≥ 2), then carry out the same check: calculate aₙ₊₁ − aₙ and see whether the result is constant and independent of n.
How do you know that a sequence is not arithmetic?
If calculating aₙ₊₁ − aₙ gives an expression that does depend on n (for example 2n + 1) — the sequence is not arithmetic, because the difference between consecutive terms changes and is not constant. Example: aₙ = n² gives aₙ₊₁ − aₙ = 2n + 1, which is not constant.
How do you tell an arithmetic sequence from a geometric one?
In an arithmetic sequence you add a fixed number each time (the common difference d), so the difference between consecutive terms is constant. In a geometric sequence you multiply by a fixed number each time (the common ratio r), so the ratio between consecutive terms is constant. Quick test: take three consecutive terms — if a₂ − a₁ = a₃ − a₂ it is arithmetic, and if a₂/a₁ = a₃/a₂ it is geometric. Example: 2, 4, 6 is arithmetic (d = 2), but 2, 4, 8 is geometric (r = 2).
Why does (n − 1) appear in the nth-term formula and not n?
Because from the first term to the nth term you take n − 1 steps, not n. To get from a₁ to a₅ in an arithmetic sequence you add the difference 4 times (a₅ = a₁ + 4d), and to get to a₅ in a geometric sequence you multiply by the ratio 4 times (a₅ = a₁·r⁴). This is the most common mistake in the topic — always count steps between positions, not the number of terms.
When does a geometric series have a sum to infinity?
Only when the ratio satisfies −1 < r < 1, that is |r| < 1. In that case the terms get smaller fast enough and the sum converges to the value S∞ = a₁/(1 − r). If |r| ≥ 1 the series diverges and has no finite sum. In the exam you must state explicitly that the convergence condition holds before using the formula.
How is compound interest related to sequences?
Compound interest is a geometric sequence. A sum invested at an annual rate of r percent grows every year by a constant factor 1 + r/100, and after n years equals P·(1 + r/100)ⁿ. That is exactly the nth-term formula of a geometric sequence. Simple interest (a fixed addition every year), on the other hand, is an arithmetic sequence — and hardly ever appears at A level.
Which formulae must you know by heart for a sequences question?
Five formulae: for an arithmetic sequence — the nth term aₙ = a₁ + (n − 1)·d and the sum Sₙ = n(a₁ + aₙ)/2; for a geometric sequence — the nth term aₙ = a₁·r^(n−1) and the finite sum Sₙ = a₁(rⁿ − 1)/(r − 1); and for the sum to infinity (when |r| < 1) — S∞ = a₁/(1 − r). The sums are in the A-level formula booklet, but with these five and correct identification of the type of sequence you can solve any sequences question on the Pure paper.
A graded set of exercises — from basic nth-term work to geometric series and compound interest
Practise sequences ←