Patterns

Patterns is the doorway into the sequences-and-series block of Paper One. It has appeared in 21 questions over the last 12 years — and every single one of those questions was shared with sequences and series. Pattern-spotting is how those questions start.

The topic itself is short and approachable: recognising how a list of numbers is growing, describing that growth with an expression, and using the expression to predict later values. But its real importance is what it unlocks — the arithmetic and geometric sequences of Sequences and Series are formalised patterns, so the work you do here is the first half of one of the most reliable question types on the paper.

The topic starts with the concept: what a pattern actually is — a list of values growing according to a rule. The first rule you meet is the simplest: a linear pattern, where the same amount is added each time, so the difference between terms is constant.

From there the growth rules get richer. In a quadratic pattern the difference itself changes — but at a constant rate, which links back to the quadratic expressions you met in Algebra 2. In an exponential pattern each term is multiplied by the same amount, the constant-ratio growth you know from Exponentials and Logarithms.

Then comes evaluating patterns: once a pattern is written as an expression in \(n\) — a formula for the \(n\)th term — finding any value is just substitution. Finally, applications of patterns puts the machinery into real contexts: population growth, finances, and other quantities that grow by a rule.

Patterns appears 21 times over 12 years, and its correlation table has one headline: all 21 of those questions were shared with sequences and series. Algebra and induction follow far behind on 4 each.

A note on the counts: "Algebra" bundles Algebra 1 and Algebra 2 together, and "Differentiation" bundles Differentiation 1, 2, and 3. The usual caveat — topics appearing in the same question does not always mean a deep mathematical link — barely applies here, because the link genuinely is deep: examiners treat the two topics as one continuous story. A question typically opens by showing you a pattern and asking you to describe it, then moves into sequences and series territory for the remaining parts.

Patterns sits on a short, clean chain: two topics feed in, and one topic flows out.

And outward, one arrow — but it is a big one.

There are five sub-topics in Patterns, and the order matters: the three pattern types come first, then the skills that use them. Click each step to see how they build.