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)7V(x) = \left(2 + \frac{kx}{4}\right)^7V(x)=(2+4kx)7
where 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.