For students and parents · O-Level A-Math
Number Patterns: Turning a Sequence Into a Formula
Turning a sequence into a formula for the nth term.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
A number pattern question almost never wants the next term; it wants the formula for the nth term, so that you can find the 100th term without writing out the first 99. The way in is not to stare at the numbers, it is to organise them, and the differences between terms tell you what kind of formula you are looking for.
Constant first difference means a linear formula
Take the sequence 5, 8, 11, 14. The differences are 3, 3, 3, a constant. A constant first difference means the formula is linear, of the form an + b, and the common difference is the a.
So here a = 3, and the formula is 3n + something. Test n = 1: 3(1) = 3, but the first term is 5, so add 2. The formula is 3n + 2. Check: n = 1, 2, 3, 4 give 5, 8, 11, 14. It matches, so the formula is right.
Constant second difference means a quadratic formula
Take 2, 6, 12, 20. The first differences are 4, 6, 8, not constant, so it is not linear. The second differences are 2, 2, a constant, which means the formula is quadratic, of the form an2 + bn + c. A constant second difference of 2a tells you a: here 2a = 2, so a = 1.
With a = 1, subtract n2 from each term: 2 - 1 = 1, 6 - 4 = 2, 12 - 9 = 3, 20 - 16 = 4, which is the sequence 1, 2, 3, 4, that is just n. So the formula is n2 + n. Check: n = 1, 2, 3, 4 give 2, 6, 12, 20. It matches.
The routine
Find the first differences. If they are constant, the formula is linear and the common difference is the coefficient of n. If they are not, find the second differences; if those are constant, the formula is quadratic and half the second difference is the coefficient of n2. Then peel off that leading part and read what is left. Organising the differences is what makes the formula appear; staring at the original numbers does not.
The teaching point
Patterns are solved by organising, not by staring. A constant first difference points to a linear formula, a constant second difference to a quadratic one, and from there the formula is built one layer at a time and then checked against the given terms. That habit, represent then read, carries from PSLE number patterns straight into A-Math sequences.
If your child can spot the next term but cannot reach the formula, that is the exact step GPA teaches, on questions from PSLE through O-Level. A short diagnostic consult will show where the method stalls.
Both nth-term formulas on this page were checked against the sequences before publishing.