Sequences and the nth term
Maths · Key Stage 3 · The School
Find the step
For 5, 8, 11, 14… the step is +3 every time. A constant step means the sequence is LINEAR, and its nth term is always (step × n) + adjustment. Here 3n gives 3, 6, 9, 12 — each is 2 short of our sequence, so the nth term is 3n + 2. Check with n = 4: 3(4) + 2 = 14. ✓
Why the nth term is worth having
It jumps straight to any term: the 100th term of 3n + 2 is 302, with no listing. It also answers "is 89 in this sequence?" — solve 3n + 2 = 89, n = 29, a whole number, so yes, it is the 29th term. If n came out as 29.6, the answer would be no. That is a proper mathematical argument, not a guess.
The famous non-linear ones
Squares 1, 4, 9, 16 (n²). Cubes 1, 8, 27, 64 (n³). Triangular 1, 3, 6, 10 — the steps grow by one each time, nth term n(n+1)/2. Fibonacci 1, 1, 2, 3, 5, 8 — each term is the sum of the two before, so it has no simple nth-term formula. When the step is not constant, check the SECOND differences: constant second differences mean a quadratic.
The step is +3 each time, so the rule starts 3n. Test it: 3n gives 3, 6, 9, 12. Compare with the sequence: 5, 8, 11, 14. Every term is 2 bigger, so the nth term is 3n + 2. Check at n = 1: 3 + 2 = 5. ✓ Check at n = 4: 12 + 2 = 14. ✓ Now use it: the 50th term is 150 + 2 = 152. And is 89 in this sequence? 3n + 2 = 89 gives n = 29, a whole number, so yes — it is the 29th term.