Sequences (GCSE)

Linear, quadratic, geometric, Fibonacci-like. Higher tier requires quadratic n^th term.

Linear (KS3): a×n + b. Quadratic (Higher): an² + bn + c, with a found from half the second difference. Geometric: fixed ratio between terms.

Worked examples

1, 4, 7, 10, ... linear: 3n − 2.

2, 5, 10, 17, 26, ... quadratic: n² + 1. Differences 3, 5, 7, 9; second difference 2 → a = 1.

2, 6, 18, 54, ... geometric: ratio 3. n^th term = 2 × 3ⁿ⁻¹.

Frequently asked questions

Quadratic n<sup>th</sup> term method?

Find a from second difference / 2. Subtract a×n² from the original; the remainder is a linear sequence whose n^th term is bn + c.

Fibonacci?

Each term is the sum of the previous two. 1, 1, 2, 3, 5, 8, 13. Recursive rather than closed-form.