The volume of a growing tropical fruit, V(x)V(x)V(x) in cm3\text{cm}^3cm3, after x x\,x weeks is modeled by the function
V(x)=(2+kx4)7 V(x) = \left(2 + \frac{kx}{4}\right)^7 V(x)=(2+4kx)7where k k\,k is a non-zero constant.
Find the first 4 terms, in ascending powers of xxx, of the binomial expansion of V(x)V(x)V(x). Give each term in simplest form.
Given that the coefficients of xxx, x2x^2x2, and x3 x^3\,x3 in this expansion form the first three terms of an arithmetic progression,
find, using algebra, the possible values of kkk.
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.