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

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

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

form a geometric sequence and state the common ratio.

It is given that

$$\frac{85x}{(1+3x)(5-2x)}\equiv \frac{P}{5-2x}+\frac{Q}{1+3x}$$

where $P$ and $Q$ are integers. Find the value of $P$ and the value of $Q$.

(i) In a structural model, the stress $S$ on a beam at distance $x$ from the support is given by $S(x)=\frac{17x}{(1+3x)(5-2x)}$. Using your answers to parts (a) and (b), find the binomial expansion of $S(x)$ up to and including the term in $x^2$.

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