In a biological study, the population density P P\,P of a specific bacterial colony is modeled by the function P(c)=(4+kc)7P(c) = (4 + kc)^7P(c)=(4+kc)7, where c c\,c represents the concentration of a growth medium and k k\,k is a positive constant.
Find the first three terms, in ascending powers of ccc, in the expansion of
(4+kc)7 (4 + kc)^7 (4+kc)7Given that the coefficient of c2 c^2\,c2 in this expansion is exactly half the coefficient of ccc, determine the value of the constant 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.