An engineering model for the structural density ρ \rho\,ρ of a composite material, relative to a depth xxx, is given by the binomial expansion of (3+kx)6(3 + kx)^6(3+kx)6 in ascending powers of xxx. The expansion is written in the form:
ρ=c0+c1x+c2x2+c3x3+… \rho = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \dots ρ=c0+c1x+c2x2+c3x3+…where c0,c1,c2, c_0, c_1, c_2,\,c0,c1,c2, and c3 c_3\,c3 are constants.
Find the value of c0c_0c0.
Given that:
Find:
(i) the value of kkk
(ii) the value of c2c_2c2
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.