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 OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.