Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xₙ₊₁ = f(xₙ); increasing sequences; decreasing sequences; periodic sequences

Sequences are ordered lists of numbers that follow a specific rule. In A-Level Mathematics, you will encounter sequences defined by a formula for the nth term, such as arithmetic and geometric sequences, as well as those generated by recurrence relations like xₙ₊₁ = f(xₙ). Understanding sequences is fundamental to calculus, series, and mathematical modelling, as they appear in topics such as limits, numerical methods, and financial mathematics.

You will learn to classify sequences as increasing, decreasing, or periodic. An increasing sequence has each term greater than the previous (e.g., 2, 4, 6, 8...), while a decreasing sequence has each term smaller (e.g., 9, 6, 3, 0...). Periodic sequences repeat after a fixed number of terms, like 1, 2, 1, 2, 1, 2... . Recognising these properties helps in analysing convergence and behaviour of sequences, which is crucial for understanding series and iteration.

This topic builds on GCSE algebra and introduces notation that is used throughout A-Level. Mastery of sequences will prepare you for more advanced concepts such as recurrence relations in differential equations, convergence tests for series, and modelling real-world phenomena like population growth or radioactive decay.