Find the first four terms, in ascending powers of xxx, of the binomial expansion of (1+2x)12\displaystyle (1+2x)^{\frac12}(1+2x)21.
Hence find the expansion of
1+2x1−x\displaystyle \frac{\sqrt{1+2x}}{1-x}1−x1+2x
up to and including the term in x3x^3x3.
State the range of values of x x\,x for which both expansions used in part (b) are valid.
In this question you must show detailed reasoning.
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.