Arithmetic and geometric sequences are a classic, rewarding topic in maths exams: anyone who knows the formulae and when to use each one collects safe marks. In this guide we set the two types of sequence side by side — the nth term, the sum of the series, finding the common difference and ratio — see full examples, solve typical worded problems, and map the common mistakes, with a practice plan that will take you to full command.
Sequences are one of the friendliest topics in maths exams: there are few formulae, the sum formulae are given on the A-level formula booklet, and the questions repeat in familiar patterns. And yet many students lose marks on it — not because the topic is hard, but because they confuse an arithmetic sequence with a geometric one, or miscount the number of terms. This guide sets the two types of sequence side by side, explains every formula with an example, and shows how to identify quickly which sequence you are dealing with and which formula to use. The nth term of both types is GCSE Higher content (A24-A25); the sum of a series is A level.
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, the second a₂, and so on; the nth term is called the general term and written aₙ. The two types of sequence studied for the exams differ in how you move from one term to the next: in an arithmetic sequence you add the same number each time (a constant difference), and in a geometric sequence you multiply by the same number each time (a constant ratio).
Arithmetic sequences — formulae and examples
In an arithmetic sequence you move from one term to the next by adding a fixed number called the common difference, written d. You calculate it as: d = a₂ − a₁ (any two consecutive terms). The nth-term formula is aₙ = a₁ + (n − 1)·d. In words: start from the first term, and add the difference (n − 1) times to reach the nth term. At GCSE you will often see this written as a linear expression in n — for example 3n + 2 — which is the same formula tidied up.
Example: in the sequence 5, 8, 11, 14, … what is the 20th term? Here a₁ = 5 and d = 3. So a₂₀ = 5 + (20 − 1)·3 = 5 + 19·3 = 5 + 57 = 62. Note the multiplier (n − 1) and not n — a common mistake. To reach the 20th term you add the difference 19 times, not 20.
The sum of the first n terms of an arithmetic sequence is calculated with the formula Sₙ = n·(a₁ + aₙ)/2 — the average of the first and last terms, times the number of terms. An equivalent version: Sₙ = n·[2a₁ + (n − 1)d]/2, useful when you do not know aₙ. Example: the sum of the first 20 terms of the sequence above: S₂₀ = 20·(5 + 62)/2 = 20·67/2 = 670.
| Arithmetic sequence | Formula | Explanation |
|---|---|---|
| Common difference | d = a₂ − a₁ | The difference between two consecutive terms |
| nth term | aₙ = a₁ + (n − 1)·d | Add d exactly (n − 1) times |
| Sum (with aₙ) | Sₙ = n·(a₁ + aₙ)/2 | Average of first and last, times n |
| Sum (without aₙ) | Sₙ = n·[2a₁ + (n − 1)d]/2 | When you do not know aₙ |
Geometric sequences — formulae and examples
In a geometric sequence you move from one term to the next by multiplying by a fixed number called the common ratio, written r. You calculate it as: r = a₂ / a₁ (divide two consecutive terms). The nth-term formula is aₙ = a₁ · r^(n−1). In words: start from the first term, and multiply by the ratio (n − 1) times.
Example: in the sequence 3, 6, 12, 24, … what is the 7th term? Here a₁ = 3 and r = 2. So a₇ = 3 · 2^(7−1) = 3 · 2⁶ = 3 · 64 = 192. Again, the power is (n − 1) = 6 and not 7.
The sum of the first n terms of a geometric sequence (when r ≠ 1) is calculated with the formula Sₙ = a₁·(rⁿ − 1)/(r − 1). Example: the sum of the first 7 terms of the sequence above: S₇ = 3·(2⁷ − 1)/(2 − 1) = 3·(128 − 1)/1 = 3·127 = 381. When r is between 0 and 1 (a decreasing sequence) it is convenient to write the formula as Sₙ = a₁·(1 − rⁿ)/(1 − r) so that the numerator and denominator are positive.
| Geometric sequence | Formula | Explanation |
|---|---|---|
| Common ratio | r = a₂ / a₁ | The ratio between two consecutive terms |
| nth term | aₙ = a₁ · r^(n−1) | Multiply by r exactly (n − 1) times |
| Sum (r ≠ 1) | Sₙ = a₁·(rⁿ − 1)/(r − 1) | Sum of the first n terms |
Techniques for common problems
A common type of question: two non-consecutive terms are given, find the sequence. For example, in an arithmetic sequence a₃ = 10 and a₇ = 26. Build two equations: a₁ + 2d = 10 and a₁ + 6d = 26. Subtract: 4d = 16, so d = 4, and then a₁ = 10 − 8 = 2. Now you know the whole sequence. The same technique works for a geometric sequence, except that you divide the equations instead of subtracting to extract r.
A second type: the sum is given, find the number of terms n. Here you substitute into the sum formula and solve an equation in n (sometimes a quadratic). Example: in an arithmetic sequence a₁ = 2, d = 3, and it is given that Sₙ = 155. Substitute: n·[2·2 + (n − 1)·3]/2 = 155, that is n·(3n + 1)/2 = 155, giving 3n² + n − 310 = 0, whose positive solution is n = 10. The negative solution is rejected — the number of terms must be a natural number.
Worded problems: many of them turn an everyday situation into a sequence. A fixed monthly increase in savings = an arithmetic sequence; interest or growth by a fixed percentage = a geometric sequence. Example: someone saves £100 in the first month, and £50 more each month — that is an arithmetic sequence with a₁ = 100, d = 50. How much have they saved after a year? S₁₂ = 12·[2·100 + 11·50]/2 = 12·(200 + 550)/2 = 12·375 = £4,500.
Common mistakes that lose marks
Mistake 1 — using n instead of (n − 1) in the nth term. To reach the nth term you move (n − 1) steps from the first term, not n. This is the most common mistake in sequences. Mistake 2 — confusing the types of sequence: using the arithmetic sum formula on a geometric sequence or the other way round. Always identify the type of sequence first.
Mistake 3 — a wrong sign or order in the geometric sum formula. The formula is a₁·(rⁿ − 1)/(r − 1); when the ratio is less than 1 it is convenient to flip the order to (1 − rⁿ)/(1 − r). Mistake 4 — getting a non-integer or negative n and writing it as the answer. The number of terms must be a natural number, so a non-integer or negative solution is rejected. Mistake 5 — calculating r from non-consecutive terms without a root: if a₁ and a₃ are given in a geometric sequence, then a₃ = a₁·r², that is r² = a₃/a₁, and you must take a square root (and watch out for the two possible signs).
A sequences practice plan
Sequences are a formula-based topic, so focused practice brings quick command. The following plan assumes about an hour a day and is built for about a week and a half.
- Days 1-2: arithmetic sequences — identification, finding d, and the nth term. Twenty-five exercises, with the emphasis on (n − 1).
- Days 3-4: the sum of an arithmetic series, including finding n from the sum. Twenty exercises.
- Days 5-6: geometric sequences — finding r, the nth term and the sum. Twenty exercises.
- Day 7: problems with two given terms (simultaneous equations), in both types of sequence.
- Days 8-9: worded problems and full past-paper questions, under timed conditions.
Summary
Sequences are a topic that rewards anyone who practises: few formulae, recurring patterns, and safe marks. The key is to identify quickly whether the sequence is arithmetic (constant difference) or geometric (constant ratio), and then apply the right formula — the nth term with the multiplier or power (n − 1), and the sum by the matching formula. If you remember the (n − 1), identify the type of sequence correctly, and reject non-integer solutions for n — the sequences question in the exam will become one of your easiest and safest. The way there is through repeated practice, and that is exactly what is waiting for you on the MathsUK practice pages.
Frequently asked questions
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence you move from one term to the next by adding a constant difference d, for example 3, 7, 11, 15 with d = 4. In a geometric sequence you move by multiplying by a constant ratio r, for example 3, 6, 12, 24 with r = 2. To identify: if the difference between consecutive terms is constant — arithmetic; if the ratio between consecutive terms is constant — geometric.
What is the nth-term formula?
In an arithmetic sequence: aₙ = a₁ + (n − 1)·d. In a geometric sequence: aₙ = a₁ · r^(n−1). In both, the multiplier or power is (n − 1) and not n, because to reach the nth term you move (n − 1) steps from the first term. This is one of the points students get wrong most often.
How do you calculate the sum of a series?
For an arithmetic series: Sₙ = n·(a₁ + aₙ)/2, or Sₙ = n·[2a₁ + (n − 1)d]/2 when you do not know aₙ. For a geometric series (r ≠ 1): Sₙ = a₁·(rⁿ − 1)/(r − 1). When the ratio is less than 1 it is convenient to write the geometric formula as a₁·(1 − rⁿ)/(1 − r) so that the numerator and denominator are positive. Sums of series are A-level content; at GCSE you only need the nth term.
How do you find a sequence when two non-consecutive terms are given?
Build two equations from the nth-term formula. For an arithmetic sequence, for example a₃ = 10 and a₇ = 26 give a₁ + 2d = 10 and a₁ + 6d = 26; subtract to get 4d = 16, that is d = 4, and then a₁ = 2. For a geometric sequence do the same but divide the equations by each other to extract r (and watch out for the two possible signs when taking a root).
What is the most common mistake in sequences questions?
Using n instead of (n − 1) in the nth-term formula, and confusing an arithmetic sequence with a geometric one (using the wrong formula). Further mistakes: a wrong sign in the geometric sum formula, and writing a number of terms n that is not a positive integer — the number of terms must be a natural number, so such a solution is rejected.
How do you practise the nth term of a geometric sequence without getting confused?
The safest way is to practise in pairs: first find r from two given terms (r = a₂/a₁, or compare powers when the terms are not consecutive), and only then substitute into the formula aₙ = a₁·r^(n−1). Practise at least 15-20 exercises in a row so that counting the (n − 1) becomes automatic, not just memorising the formula by heart.
How many exercises should you do to master sequences?
A rule of thumb: around 40-50 exercises spread over about a week and a half — 20-25 on arithmetic sequences and another 20-25 on geometric sequences, including sum exercises and worded problems. The full practice plan in the guide is built exactly to that pace, day by day.
A graded set of exercises — from the nth term to the sum of a series and worded problems in exam style
Practise arithmetic and geometric sequences ←