A quantitative analyst models the projected value of a portfolio using the function
V(x)=(4+kx)7V(x) = (4 + kx)^7V(x)=(4+kx)7
where x x\,x represents market volatility and k k\,k is a positive constant.
Find the first three terms, in ascending powers of xxx, in the expansion of (4+kx)7(4 + kx)^7(4+kx)7, giving each term in its simplest form.
Given that the coefficient of x2 x^2\,x2 is exactly three-eighths of the coefficient of xxx, determine the value of kkk.
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.