The reaction rate R R\,R of a biochemical catalyst depends on its concentration ccc (in mol dm−3^{-3}−3) according to the model
R(c)=1+8c23,∣c∣<18 R(c) = \sqrt[3]{1 + 8c^2}, \quad |c| < \frac{1}{\sqrt{8}} R(c)=31+8c2,∣c∣<81Find, in ascending powers of ccc, the first three non-zero terms of the binomial series expansion of R(c)R(c)R(c), giving each coefficient as a simplified fraction.
By substituting c=14\displaystyle c = \frac{1}{4}c=41 into the expansion from part (a), find a rational approximation to k33k \sqrt[3]{3}k33, where k k\,k is a constant you must determine.
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.