Arithmetic and geometric sequences and series, sigma notation, the conditions for convergence of a geometric series, the binomial expansion for positive integer and rational powers, and recurrence relations.

What this dot point is asking

AQA wants you to handle arithmetic and geometric sequences and series, use sigma notation, know when a geometric series converges and find its sum to infinity, perform binomial expansions for positive integer powers and for rational and negative powers, and use recurrence relations. Geometric-series problems that combine two given facts into a quadratic, and binomial expansions for rational powers, are perennial exam favourites.

Arithmetic sequences and series

The $n$-th term is $u_n = a + (n - 1)d$, where $a$ is the first term and $d$ the common difference. The sum of the first $n$ terms is $S_n = \dfrac{n}{2}(2a + (n - 1)d)$, equivalently $\dfrac{n}{2}(a + l)$ where $l$ is the last term. Many problems give you two pieces of information (for instance a particular term and a particular sum) and ask you to set up and solve simultaneous equations in $a$ and $d$.

Geometric sequences and series

The $n$-th term is $u_n = ar^{n-1}$, with common ratio $r$. The sum of the first $n$ terms is $S_n = \dfrac{a(1 - r^n)}{1 - r}$ (equivalently $\dfrac{a(r^n - 1)}{r - 1}$, more convenient when $r > 1$).

A common structure combines $ar =$ (a given term) with $\frac{a}{1 - r} =$ (a given sum). Eliminating $a$ produces a quadratic in $r$, as in the worked exam question above, so expect to factorise or use the formula.

Sigma notation

The symbol $\sum$ compactly represents a sum. For example $\displaystyle\sum_{r=1}^{4}(2r + 1) = 3 + 5 + 7 + 9 = 24$. Standard results, such as $\sum_{r=1}^{n} 1 = n$ and $\sum_{r=1}^{n} c = cn$ for a constant $c$, let you split and simplify sigma expressions before evaluating.

The binomial expansion

For a positive integer $n$, $(a + b)^n = \displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$, where $\binom{n}{r} = \dfrac{n!}{r!\,(n - r)!}$. This is a finite expansion with $n + 1$ terms.

Recurrence relations

A recurrence relation defines each term from the previous one, for example $u_{n+1} = 2u_n + 3$ with $u_1 = 1$. You generate terms by repeated substitution, and some sequences settle to a limit $L$ found by solving $L = 2L + 3$ (where a stable limit exists). Recurrence questions may ask for a specific term, the sum of several terms, or a limit.

Notation, periodicity and modelling

A sequence may be defined by a position-to-term rule (a formula for $u_n$ directly in terms of $n$) or by a term-to-term rule (a recurrence). You should be able to move between them where possible, and to recognise an arithmetic or geometric sequence from either form. Some recurrences produce a periodic sequence that repeats with a fixed period; for these, the sum of a large number of terms is found by summing one full period and multiplying, then adding the leftover terms. Sigma sums of periodic sequences are a recurring exam idea.

Series model real situations: arithmetic series describe quantities that change by a fixed amount each step (such as regular savings increasing by a constant), while geometric series describe repeated proportional change (such as compound interest or a bouncing ball losing a fixed fraction of height). The sum to infinity of a convergent geometric series gives the total distance a ball travels or the eventual total of an infinite repayment, but only when $|r| < 1$. Setting up the right type of series, with the correct first term and common difference or ratio, is the modelling skill these questions test.