Find the binomial expansion of (1−2x)−1(1 - 2x)^{-1}(1−2x)−1 up to and including the term in x2x^2x2.
It is given that
8x(1−2x)(3+2x)≡P1−2x+Q3+2x\displaystyle \frac{8x}{(1 - 2x)(3 + 2x)} \equiv \frac{P}{1 - 2x} + \frac{Q}{3 + 2x}(1−2x)(3+2x)8x≡1−2xP+3+2xQ
where P P\,P and Q Q\,Q are integers.
Find the value of P P\,P and the value of QQQ.
Using your answers to parts (a) and (b), find the binomial expansion of
8x(1−2x)(3+2x)\displaystyle \frac{8x}{(1 - 2x)(3 + 2x)}(1−2x)(3+2x)8x
up to and including the term in x2x^2x2.
Find the range of values of x x\,x for which the expansion in part (c) is valid.
308 exam-style questions on AQA A Level Maths 1.7 D: Sequences and series, covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.