A-Level Maths: Binomial Series and Recurrence Relations

In the second year of A level Mathematics, the binomial expansion is extended beyond positive whole-number powers, and sequences are defined by rules that generate each term from the one before. This material trains both parts of that content, based on the sequences and series section of the DfE A level Mathematics subject content.

The binomial questions use $(1+x{)}^n=1+nx+\frac{n(n-1)}{2!}{x}^2+\dots$ for negative and fractional $n$. You expand expressions such as $(1+x{)}^{-2}$, $(1-4x{)}^{\frac{1}{2}}$ and $\frac{1}{2+x}$, where a factor must be taken out first to reach the form $a^n{(1+\frac{b}{a}x)}^n$. Every expansion comes with its range of validity, $\mid \frac{bx}{a}\mid <1$, and you use expansions to approximate values such as $\sqrt{0.96}$ by choosing a small $x$ inside that range. Other questions combine an expansion with a product or with partial fractions to find a particular coefficient.

The recurrence questions use rules of the form $u_{n+1}=f(u_n)$. You generate terms, find an unknown constant from a given term, and decide whether a sequence is increasing, decreasing or periodic. A periodic sequence such as $u_{n+1}=\frac{1}{1-u_n}$ lets you find the sum of many terms by counting complete cycles. Modelling questions then use arithmetic and geometric series in context, for example a salary rising by a fixed percentage each year or savings that increase by a fixed amount each month, and ask when a total first passes a target and how realistic the model is.

All values are given in each question, and money is in pounds.

The material offers three formats. The quiz has 12 multiple-choice questions with explanations that show each expansion or recurrence step and the common slips behind the wrong options, such as forgetting to square the whole term or to multiply by $a^n$. The flashcards summarise the general binomial expansion, the validity conditions and the definitions of increasing, decreasing and periodic sequences, together with the series formulae. The written work gives 8 longer questions to answer by hand, including an expansion found through partial fractions, the limit of a recurrence and a depreciation model to evaluate.

• Expand (1 + x)^n and (a + bx)^n for negative and fractional n
• State the range of validity of a binomial expansion
• Use a binomial expansion to approximate a value
• Find coefficients in expansions of products and of expressions split into partial fractions
• Generate and analyse sequences defined by recurrence relations
• Identify increasing, decreasing and periodic sequences
• Model real situations with arithmetic and geometric series

Practice material written by Zestly, based on the DfE A level Mathematics subject content (sequences and series: binomial expansion for rational n and its validity, sequences defined by recurrence relations, increasing, decreasing and periodic sequences, arithmetic and geometric series in modelling).