Sequences: linear, quadratic and geometric
Maths · GCSE · The School
Linear sequences
For a sequence going up by the same amount each time, the common difference is the number in front of n. Write that first, list what it produces, and compare with the sequence you were given. The gap between the two is the constant you add or subtract. Two steps, and the second one is the step everybody skips.
When the difference changes
If the differences are not constant, take the differences of the differences. If THOSE are constant, the sequence is quadratic and the n squared coefficient is half that second difference. If instead each term is a fixed multiple of the one before, it is geometric — 2, 6, 18, 54 multiplies by 3 each time, and no amount of subtracting will reveal it.
Is this number in the sequence?
Set the nth-term rule equal to the number and solve for n. If n comes out a positive whole number, the number is in the sequence and n tells you its position. If n is a fraction or negative, it is not. Answering "yes" without showing that n is a whole number earns nothing, because the whole question is about that test.
The differences are 3, 3, 3 — constant, so it is linear and the rule begins 3n. WHAT DOES 3n GIVE? For n = 1, 2, 3, 4 it gives 3, 6, 9, 12. Compare with 5, 8, 11, 14: every one is 2 bigger. So the rule is 3n + 2. TEST IT: n = 1 gives 5 ✓, n = 4 gives 14 ✓. NOW USE IT. What is the 50th term? 3 × 50 + 2 = 152. Is 89 in the sequence? Solve 3n + 2 = 89, so 3n = 87 and n = 29 — a whole number, so yes, it is the 29th term. Is 90? 3n = 88, n = 29.33, not a whole number, so no. One rule, and three different questions answered from it.
Finding a quadratic nth term, worked
For 2, 7, 14, 23, 34: the first differences are 5, 7, 9, 11 and the second differences are 2, 2, 2 — constant, so it is quadratic, and half of 2 gives the n² coefficient: n². Subtract n² (1, 4, 9, 16, 25) from the sequence: 1, 3, 5, 7, 9 — that is linear with rule 2n − 1. So the full nth term is n² + 2n − 1. Test it: n = 1 gives 1 + 2 − 1 = 2 ✓, n = 4 gives 16 + 8 − 1 = 23 ✓.
The nth term of a geometric sequence
For a geometric sequence, the nth term is a × r to the power (n − 1), where a is the first term and r is the common ratio. For 3, 6, 12, 24: a = 3 and r = 2, so the nth term is 3 × 2 to the power (n − 1). The 6th term is 3 × 2⁵ = 3 × 32 = 96 — found without listing every term in between, which is the whole point of having a formula.