A-level Mathematics₇₃₅₇

Understand and use the binomial expansion of ${\mathsf{(}\mathit{a}\mathsf{+}\mathit{b}\mathit{x}\mathsf{)}}^{\mathit{n}}\mathit{ }$ for positive integer $\mathit{n}$ ; the notations $\mathit{n}\mathsf{!}$ , $\mathit{n}\mathsf{C}\mathit{r}$ and $\left(\begin{array}{c}\mathit{n}\\ \\ \mathit{r}\end{array}\right)\mathit{}$ ; link to binomial probabilities.

Extend to any rational $\mathit{n}$ , including its use for approximation; be aware that the expansion is valid for $\left|\frac{\mathit{b}\mathit{x}}{\mathit{a}}\right|\mathsf{<}\mathsf{1}$ . (Proof not required.)

Work with sequences including those given by a formula for the $\mathit{n}$ th term and those generated by a simple relation of the form ${\mathit{x}}_{\mathit{n}\mathsf{+}\mathsf{1}}\mathsf{=}\mathrm{f}\left({\mathit{x}}_{\mathit{n}}\right)$ ; increasing sequences; decreasing sequences; periodic sequences.

Understand and use sigma notation for sums of series.

Understand and work with arithmetic sequences and series, including the formulae for $\mathit{n}$ th term and the sum to $\mathit{n}$ terms.

Understand and work with geometric sequences and series including the formulae for the $\mathit{n}$ th term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of $\left|\mathit{}\mathit{r}\mathit{}\right|\mathsf{<}\mathsf{1}$ ; modulus notation.

Use sequences and series in modelling.