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 (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.