# Sequences and Series in A-Level Maths

## Sequences & Series

You have seen some **Sequences and Series** before. Recall that during GCSE Maths you were taught the nth term for linear and quadratic sequences and you also looked at compound interest.

The difference between a sequence and a series is that the terms in a sequence are listed whereas the terms in series are summed. Of course, you can have an infinite sequence but for an infinite series to exist, the summation must converge. Examples of infinite series include binomial expansions when powers of your binomial are negative or fractional or both. **Binomial Expansion** is studied in both years of A-Level Maths. Firstly, positive integer powers of the expansion are studied in Year 1 – this is the only topic we study in Sequences and Series in Year 1. Secondly, as mentioned above, negative/fractional powers are studied in Year 2. Note that the formulae for binomial expansions is given in the formula booklet:

In addition to binomial expansion with negative/fractional powers, you will study arithmetic and geometric series. In an arithmetic series the terms change by a common factor, whereas in a geometric series, they change by a common factor. You may also see closed form sequences and sigma notation during this second year.

## Other Areas in AS-Maths

- PROOF – proof by deduction, proof by exhaustion, disproof by counterexample
- ALGEBRA & FUNCTIONS – completing the square, cubics, curve sketching, discriminant, indices, inequalities, polynomials, quadratics, simultaneous equations, surds, transformations
- COORDINATE GEOMETRY – straight lines, equation of a circle
- TRIGONOMETRY – non-right-angled triangles, trigonometric equations, trigonometric graphs, trigonometric identities
- EXPONENTIALS & LOGS – exponential & logarithmic graphs, logs, their rules and solving log equations, growth & decay, differentiating e to the kx
- DIFFERENTIATION – differentiation from first principles and differentiating polynomials, increasing & decreasing functions, stationary points, tangents & normals, differentiating e to the kx
- INTEGRATION – fundamental theorem of calculus and integrating powers of x, definite integrals
- NUMERICAL METHODS – (not covered at AS Level)
- VECTORS – two-dimensional vectors, vector arithmetic, vectors in context

## Other Areas in A2-Maths

- PROOF– proof by contradiction
- ALGEBRA & FUNCTIONS – modulus of a function, partial fractions, inverse and composite functions, compound transformations
- COORDINATE GEOMETRY – parametric equations
- TRIGONOMETRY – radians, arc length & area of a sector, small angle approximations, reciprocal trigonometric functions, inverse trigonometric functions, double & compound angle formulae
- EXPONENTIALS & LOGARITHMS – compound transformations
- DIFFERENTIATION – concavity, convexity & inflection points, derivatives of trigonometric functions, product, quotient & chain rule, parametric & implicit differentiation, differentiating exp/log functions, differential equations & rates
- INTEGRATION – further integration, integration using trigonometric identities, integration by substitution, integration by parts, integration using partial fractions, solving differential equations,
- NUMERICAL METHODS – locating roots using iteration (including Newton-Raphson), trapezium rule
- VECTORS – 3D vectors