The population density ρ(r)\rho(r)ρ(r) of a certain bacterial colony as a function of the distance r r\,r from its center is modeled by the function ρ(r)=(3+2r)−3\rho(r) = (3 + 2r)^{-3}ρ(r)=(3+2r)−3 for ∣r∣<1.5|r| < 1.5∣r∣<1.5.
Find the binomial expansion of ρ(r)\rho(r)ρ(r) in ascending powers of r r\,r up to and including the term in r3r^3r3, fully simplifying each coefficient.
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.