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Sequences and Series

Arithmetic and geometric sequences and their sums: the formal machinery behind patterns, and the engine that financial maths runs on.

Topic Importance

Sequences and Series is the formal half of the patterns block on Paper One. It has appeared in 21 questions over the last 12 years — and every single one of those questions was shared with patterns. A pattern opens the question; the sequences and series machinery earns the rest of the marks.

Sequences and series appeared in 21 questions over the last 12 years, every single one shared with patterns

This is one of the most predictable question types on the paper: recognise the sequence, name its variables, and apply the right term or sum formula. And the topic pays forward — the geometric series is the engine of Financial Maths, where compound interest and loan repayment questions are geometric series in disguise.

What the Topic Covers

The topic starts with the concept: a sequence is an ordered list of numbers built by a rule, with \(T_n\) naming the \(n\)th term. The first kind you study in detail is the arithmetic sequence — the same amount added each time, a formalised linear pattern.

Then comes the idea that gives the topic its double name: a series is the sum of the terms of a sequence so far. The same list of numbers can be read two ways.

The sequence 2, 5, 8, 11 shown as terms, and beneath it the series: the running totals 2, 7, 15, 26

The arithmetic series comes with a sum formula and its own applications. Then the whole process repeats for the multiplicative type: geometric sequences — constant ratio, formalised exponential patterns — followed by the geometric series and its sum formulas.

The topic closes with behaviour in the long run: whether a sequence converges or diverges as \(n\) grows, and the surprising result that an infinite series can add up to a finite number.

Example question

The arithmetic sequence \(2,\; 5,\; 8,\; \dots\) has first term \(a = 2\) and common difference \(d = 3\). Find the sum of the first 20 terms.

Example question

Evaluate \(\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots\)

Exam Correlations

Sequences and series appears 21 times over 12 years, and the correlation table has the same headline as the Patterns article, seen from the other side: all 21 of those questions were shared with patterns. Algebra and induction follow far behind on 4 each.

Questions shared with sequences and series over 12 years: patterns 21, algebra 4, induction 4, exponentials and logs 3, and smaller counts for the remaining topics

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 sequences genuinely are formalised patterns: examiners treat the two topics as one continuous story. One number understates its link: Financial Maths shows only 2 shared questions, because financial questions are counted as their own topic — but mathematically they lean on the geometric series throughout.

Studied straight after Patterns, this topic completes one of the most predictable question types on Paper One — and hands you the machinery that Financial Maths questions are built from.

Concept Connections

Sequences and series sits in the middle of a single clean chain: one topic feeds in, and one topic flows out.

Patterns feeds into sequences and series, which leads out to financial maths

Comes from The idea it gives you Where you will use it in Sequences and Series

Patterns \(5,\; 8,\; 11,\; 14,\; \dots\) Growth rules — arithmetic sequences formalise linear patterns, and geometric sequences formalise exponential patterns. Pattern-spotting is how every question here begins.

Topic A question you will meet there The Sequences and Series skill inside it

Financial Maths Monthly repayments on a loan The geometric series — compound interest grows geometrically, and repayment questions sum a geometric series with the \(S_n\) formula.

Study Order

There are seven sub-topics in Sequences and Series, and they follow a deliberate rhythm: sequence, then series, for the arithmetic type — then the same again for the geometric type — then the long-run behaviour of both. Click each step to see how they build.

Introduction to Sequences

What a sequence is: an ordered list of numbers built by a rule, with \(T_n\) naming the \(n\)th term. The first examples are arithmetic sequences — the formalised linear patterns from the Patterns topic.

\(2,\; 5,\; 8,\; 11,\; \dots \quad T_1 = 2,\; T_2 = 5,\; \dots\)

Arithmetic Sequences

The first sequence type in detail: a first term \(a\) and a common difference \(d\) added each step. The \(n\)th-term formula turns "keep adding" into a single expression you can evaluate directly — the substitution skill from evaluating patterns.

\(T_n = a + (n-1)\,d\)

Arithmetic Series and Applications

The series idea arrives: \(S_n\) is the sum of the terms so far. For arithmetic sequences there is a closed formula, and this step also covers its applications — real quantities that accumulate by a constant amount.

\(S_n = \dfrac{n}{2}\left(2a + (n-1)\,d\right)\)

Geometric Sequences

The rhythm repeats for the multiplicative type: a first term \(a\) multiplied by a common ratio \(r\) each step — the formalised exponential pattern, leaning on the powers from Exponentials and Logarithms. You learn to find the variables \(a\) and \(r\) from given terms and to evaluate any term.

\(T_n = a\,r^{\,n-1}\)

Geometric Series and Sum Formulas

As step 3 did for arithmetic sequences, this step sums the geometric ones. This formula is the one Financial Maths will run on.

\(S_n = \dfrac{a\left(1 - r^{\,n}\right)}{1 - r}, \quad r \neq 1\)

Convergence and Divergence of Sequences

With both types mastered, the questions turn to long-run behaviour: what happens to the terms as \(n\) grows without limit. Some sequences settle towards a value — they converge — and others grow without bound and diverge. For geometric sequences the common ratio decides which.

\(|r| < 1 \;\Rightarrow\; T_n = a\,r^{\,n-1} \to 0\)

Convergence and Infinite Series

The final step puts convergence and series together: when the terms of a geometric sequence shrink fast enough, adding infinitely many of them gives a finite answer — the sum to infinity.

\(S_\infty = \dfrac{a}{1 - r}, \quad |r| < 1\)

Partial sums of two geometric series: with ratio between minus one and one the sums level off at the sum to infinity, and with ratio above one they grow without bound

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