(i) Find the binomial expansion of $(1+3p{)}^{-1}$ up to and including the term in $p^2$.

(ii) Show that the first three terms in the binomial expansion of

$$\frac{1}{2-5p}$$

form a geometric sequence and state the common ratio.

For an automated control system, the sensitivity factor $S(p)$ depends on a small perturbation $p$, where

$$S(p)=\frac{55p}{(1+3p)(2-5p)}$$

Express $S(p)$ in the form $\frac{P}{2-5p}+\frac{Q}{1+3p}$ where $P$ and $Q$ are integers.

(i) Using your answers to parts (a) and (b), find the binomial expansion of

$$G(p)=\frac{22p}{(1+3p)(2-5p)}$$

up to and including the term in $p^2$.

(ii) Determine the range of values of $p$ for which the binomial expansion of $G(p)$ is valid.