Problem

The binomial expansion of

$$(3+kx)^{-2}\qquad |kx|<3$$

where $k$ is a non-zero constant, may be written in the form

$$A+Bx+Cx^2+Dx^3+\cdots$$

where $A,B,C$ and $D$ are constants.

(a) Find the value of $A$

Given that $C=3B$

(b) show that

$$k^2+6k=0$$

(c) Hence (i) find the value of $k$

(ii) find the value of $D$