You are given that
(1+x)−12=1−12x+38x2−516x3+⋯\displaystyle (1+x)^{-\frac12}=1-\frac12x+\frac38x^2-\frac5{16}x^3+\cdots(1+x)−21=1−21x+83x2−165x3+⋯.
Find the expansion of 1+2x1+x\dfrac{1+2x}{\sqrt{1+x}}1+x1+2x up to and including the term in x3x^3x3.
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.