In a biological study, the population density P P\,P of a specific bacterial colony is modeled by the function P(c)=(4+kc)7P(c) = (4 + kc)^7P(c)=(4+kc)7, where c c\,c represents the concentration of a growth medium and k k\,k is a positive constant.
Find the first three terms, in ascending powers of ccc, in the expansion of
(4+kc)7 (4 + kc)^7 (4+kc)7Given that the coefficient of c2 c^2\,c2 in this expansion is exactly half the coefficient of ccc, determine the value of the constant kkk.
