What you'll learn
Sequences are ordered lists of numbers that follow a rule. In this guide you will learn how to recognise arithmetic and geometric sequences, how to find the nth term of an arithmetic sequence, how to continue and describe sequences, and how to use the nth term to find or test specific terms. These ideas appear in pattern, algebra and problem-solving questions at GCSE.
Key terms and definitions
Sequence — an ordered list of numbers following a rule.
Term — a number in the sequence; the nth term is the term in position n.
Arithmetic sequence — a sequence with a constant difference between terms.
Common difference — the fixed amount added (or subtracted) each time in an arithmetic sequence.
Geometric sequence — a sequence where each term is multiplied by a constant ratio.
nth term — a rule (formula) for the term in position n.
Core concepts
Arithmetic sequences
An arithmetic sequence changes by a constant difference each time. For example, 3, 7, 11, 15, … has a common difference of +4. If the difference is negative, the terms decrease (e.g. 20, 17, 14, …). Recognising the common difference is the key first step.
Finding the nth term of an arithmetic sequence
The nth term of an arithmetic sequence is given by nth term = dn + (a − d), where d is the common difference and a is the first term. In practice: the coefficient of n is the common difference, then adjust the constant to match the first term. For 3, 7, 11, 15: difference 4, so start with 4n; 4 × 1 = 4, but the first term is 3, so subtract 1 → 4n − 1.
Using the nth term
Once you have the nth term, you can find any term by substituting its position, or test whether a number is in the sequence by setting the nth term equal to it and checking n is a positive whole number. For 4n − 1, the 10th term is 4(10) − 1 = 39.
Geometric sequences
A geometric sequence multiplies by a constant ratio each time. For example, 2, 6, 18, 54, … multiplies by 3 each time (common ratio 3). Other examples include 80, 40, 20, 10, … (ratio ½). Identify the ratio by dividing a term by the previous term.
Other sequences
Some sequences follow other patterns: square numbers (1, 4, 9, 16, …), cube numbers (1, 8, 27, …), triangular numbers (1, 3, 6, 10, …), and the Fibonacci-type sequences where each term is the sum of the two before. Recognising these helps you continue and describe them.
Worked examples
Example 1: nth term
Find the nth term of 5, 8, 11, 14, …
Common difference 3, so start with 3n; 3 × 1 = 3 but the first term is 5, so add 2: 3n + 2.
Example 2: Is a number in the sequence?
Is 100 a term of 3n + 2?
Set 3n + 2 = 100 → 3n = 98 → n = 32.67. Not a whole number, so 100 is not a term.
Example 3: Geometric sequence
Find the next term of 4, 12, 36, …
Common ratio = 12 ÷ 4 = 3, so next term = 36 × 3 = 108.
Common mistakes and how to avoid them
Using the first term as the coefficient of n. The coefficient is the common difference, not the first term.
Sign errors with decreasing sequences. A negative difference gives a negative coefficient of n.
Confusing arithmetic and geometric. Arithmetic adds a constant; geometric multiplies by a constant.
Accepting non-integer n. A number is only in the sequence if n is a positive whole number.
Forgetting to adjust the constant. After using the difference for dn, adjust to match the first term.
Exam technique for Sequences
Find the common difference or ratio first.
Build the nth term as (difference)n, then adjust the constant.
Substitute positions to find specific terms.
Solve for n to test membership, checking n is a positive integer.
Recognise special sequences (squares, cubes, triangular, Fibonacci).
Quick revision summary
A sequence follows a rule. An arithmetic sequence has a constant common difference (3, 7, 11, … differ by 4). Its nth term uses the difference as the coefficient of n, then adjusts the constant to match the first term (3, 7, 11, 15 → 4n − 1). Use the nth term to find any term (substitute n) or to test membership (set it equal to the number and check n is a positive whole number). A geometric sequence multiplies by a constant ratio each time (2, 6, 18 → ratio 3), found by dividing consecutive terms. Other patterns include square, cube, triangular and Fibonacci-type sequences. The main mistakes are using the first term instead of the difference as the n coefficient, sign errors in decreasing sequences, confusing arithmetic with geometric, and accepting non-integer n. Find the difference or ratio, build and adjust the nth term, substitute to find terms, and solve for n to test membership — these steps handle the great majority of sequence questions.