Sequences
Term-to-term and position-to-term rules, including the nth term.
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.6.1Continue sequences and find the nth term of linear and simple quadratic sequences.
5 minute read
Sequences
Linear (arithmetic) sequences
A sequence with a constant difference between terms. To find the nth term:
- Find the common difference d. This is the coefficient of n.
- Work out the zero-th term — what would come before the first term.
- The nth term is
dn + (zero-th term).
For 5, 8, 11, 14: d = 3, and the term before 5 would be 2, so the nth term is 3n + 2. Check with n = 1: 3(1) + 2 = 5 ✓
Why the "zero-th term" method works
Writing 3n alone gives 3, 6, 9, 12 — each term is 2 less than the sequence we want, so we add 2. Understanding this makes the method memorable rather than mechanical.
Quadratic sequences
If the first differences are not constant, look at the second differences. If those are constant, the sequence is quadratic and the coefficient of n² is half the second difference. For 3, 8, 15, 24: first differences 5, 7, 9; second difference 2, so the n² coefficient is 1. Subtracting n² (1, 4, 9, 16) from the sequence leaves 2, 4, 6, 8 — which is 2n. So the nth term is n² + 2n.
Other sequences worth recognising
Square numbers (n²), cube numbers (n³), triangular numbers, and geometric sequences where each term is multiplied by a constant.
Think of it like this
The nth term is a formula that jumps straight to any position, like a page number rather than turning pages one at a time. That is exactly why it beats the term-to-term rule when asked for the 100th term.
Worked examples
Method, step by step
Find the nth term of the sequence 7, 12, 17, 22, ...
- 1The common difference is 5, so the rule begins 5n.
- 25n alone gives 5, 10, 15, 20 — each is 2 less than the sequence.
- 3So add 2: the nth term is 5n + 2.
- 4Check n = 1: 5(1) + 2 = 7 ✓ and n = 4: 5(4) + 2 = 22 ✓
5n + 2
Common misconceptions
- Writing the nth term as "+3" or "3n" alone. A rule must let you calculate any term, so it needs both the difference and the constant.
- Assuming every sequence is linear. If first differences vary, check second differences before forcing a linear rule.
- Testing the formula on no terms. Substituting n = 1 takes seconds and catches a wrong constant immediately.
In the exam
- Always verify your nth term by substituting n = 1 and n = 2 and checking against the given sequence.
- For quadratic sequences, remember the n² coefficient is **half** the constant second difference — not the second difference itself.