In the calibration of a precision acoustic sensor, the signal response R(s)R(s)R(s) for a sensitivity parameter s s\,s is modeled by the function
R(s)=(2+ks4)5 R(s) = \left( 2 + \frac{ks}{4} \right)^5 R(s)=(2+4ks)5where k k\,k is a non-zero constant.
Find the first 4 terms, in ascending powers of sss, of the binomial expansion of R(s)R(s)R(s). Give each term in its simplest form.
An enhanced signal model is defined by
G(s)=(5+2s)(2+ks4)5 G(s) = (5 + 2s) \left( 2 + \frac{ks}{4} \right)^5 G(s)=(5+2s)(2+4ks)5In the series expansion of G(s)G(s)G(s), the coefficient of s s\,s is equal to the constant term. Find the value of kkk.
