Sequences

A sequence is a list of numbers that follows a rule, such as 7, 11, 15, 19, ... Each number is a term. You can describe a sequence with a term-to-term rule (here, "add 4") or a position-to-term rule called the nth term (here, 4n + 3). The main types are arithmetic (add the same amount each time), geometric (multiply by the same amount), quadratic, and Fibonacci-type sequences, where each term comes from the ones before.

Term-to-term and position-to-term rules

AQA GCSE Maths asks you to generate terms from either kind of rule (A23):

• Term-to-term rule: tells you how to get from one term to the next. For 7, 11, 15, 19, the rule is "add 4". You need the first term to start.
• Position-to-term rule (nth term): a formula in n, the position in the sequence. For 4n + 3, the 1st term is 4(1) + 3 = 7, the 2nd is 11, and the 100th is 4(100) + 3 = 403, without listing all the terms before it.

Types of sequences

Sequences can also come from patterns. The triangular numbers count the dots in growing triangles:

Arithmetic sequences and the nth term

In an arithmetic sequence the difference between terms is always the same. Its nth term has the form dn + c, where d is the common difference.

Example 1: find the nth term of 7, 11, 15, 19, ...

1. The common difference is 4, so the nth term starts 4n.
2. The 4 times table is 4, 8, 12, 16. Each term of the sequence is 3 more.
3. So the nth term is 4n + 3. Check: when n = 3, 4(3) + 3 = 15.

Example 2: a decreasing sequence

For 20, 17, 14, 11, ... the difference is −3, so start with −3n: −3, −6, −9. Each term is 23 more, so the nth term is 23 − 3n.

Example 3: is a number in the sequence?

Is 251 a term of 4n + 3? Solve 4n + 3 = 251: 4n = 248, n = 62. The answer is a whole number, so 251 is the 62nd term. If n came out as a decimal, the number would not be in the sequence.

For sequences with a constant second difference, see quadratic sequences.

Geometric sequences

In a geometric sequence each term is the previous one multiplied by the common ratio r. With first term a, the nth term is:

For 3, 6, 12, 24, ..., a = 3 and r = 2, so the 10th term is 3 × 2⁹ = 1536. A ratio between 0 and 1 makes the terms shrink, as in 24, 12, 6, 3, ... where r = ½.

Here x is the position n and y is the term. The terms of a geometric sequence lie on an exponential curve, while the terms of an arithmetic sequence lie on a straight line. You can generate both and see their differences or ratios in the Sequences and Series calculator.

Check your understanding.

Sequences at A-level

Recurrence relations

AQA A-level Maths works with sequences given by an nth term formula and by a relation of the form u_n+1 = f(u_n), and classifies them as increasing, decreasing or periodic (D2).

• Increasing: u_n+1 = 2u_n − 3 with u₁ = 4 gives 4, 5, 7, 11, 19, ... Each term is bigger than the one before.
• Decreasing: 24, 12, 6, 3, ... Each term is smaller than the one before.
• Periodic: u_n+1 = 1 ÷ (1 − u_n) with u₁ = 2 gives 2, −1, ½, 2, −1, ½, ... The terms repeat every 3 steps, so its period is 3.

Series: adding up the terms

A series is the sum of the terms of a sequence, often written with sigma notation (D3). For an arithmetic series with first term a and common difference d, and for a geometric series with |r| < 1:

The first 20 terms of 7, 11, 15, ... add up to 20 ÷ 2 × (14 + 19 × 4) = 900. The geometric series 24 + 12 + 6 + 3 + ... never stops, but its sum gets closer and closer to 24 ÷ (1 − ½) = 48, its sum to infinity. Our guides to the arithmetic sequence calculator and the geometric series calculator go further.

Common mistakes

• Writing the term-to-term rule as the nth term. For 7, 11, 15, the nth term is 4n + 3, not n + 4.
• Getting the sign wrong in a decreasing sequence. A difference of −3 gives −3n.
• Using rⁿ instead of rⁿ⁻¹ in a geometric nth term.
• Saying a number is in a sequence when solving gives a decimal n. Only whole number positions count.
• Using the sum to infinity when |r| ≥ 1. The series only converges when −1 < r < 1.

Practise with a short timed paper

Sequences come up in GCSE Maths on both tiers and in A-level Maths. The daily GCSE Maths Foundation, GCSE Maths Higher and A-level Maths: Algebra & Functions papers are short timed papers of 8 questions, with a new one every day.

Read next

Maths · AlgebraQuadratic sequences: how to find the nth termHow to spot a quadratic sequence and find its nth term an² + bn + c, step by step, with worked GCSE examples, a check method and practice questions.