Describing a sequence
A sequence is an ordered list of numbers (terms) that follow a rule. A term-to-term rule tells you how to get from one term to the next (e.g. "add 3"). A position-to-term rule (the nth term) gives any term directly from its position n.
Key idea
For a linear (arithmetic) sequence with common difference d and first term a, the nth term = a + (n − 1)d, often written as dn + c. A geometric sequence multiplies by a constant ratio each time.
Finding the nth term
Worked example
Find the nth term of 5, 8, 11, 14, … .
The common difference is d = 3, so the rule starts 3n.
Compare: 3 × 1 = 3, but the first term is 5, so add 2.
nth term = 3n + 2.
Check: n = 4 gives 3(4) + 2 = 14. ✓ The 10th term = 3(10) + 2 = 32.
Special sequences
Learn to recognise: square numbers 1, 4, 9, 16, … (nth term n²); cube numbers 1, 8, 27, …; and the Fibonacci sequence 1, 1, 2, 3, 5, 8, … where each term is the sum of the previous two.
Remember
- Constant difference → linear, use dn + c.
- Constant ratio → geometric.
- Always check your rule with a known term.