#### Answer

$a_5=-729 \text{ and } a_n=-9(-3)^{n-1}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find $a_5$ and $a_n$ for the given geometric sequence described as \begin{array}{l}\require{cancel} a_4=243 \\ r=-3 ,\end{array} use the formula for finding the $n^{th}$ term of a geometric sequence.
$\bf{\text{Solution Details:}}$
Using the formula for finding the $n^{th}$ term of a geometric sequence, which is given by $a_n=a_1r^{n-1},$ then \begin{array}{l}\require{cancel} a_n=a_1r^{n-1} \\\\ a_4=a_1r^{4-1} \\\\ a_4=a_1r^{3} .\end{array}
Subtituting the given values, $a_4=243$ and $r=-3,$ the equation above results to \begin{array}{l}\require{cancel} 243=a_1(-3)^{3} \\\\ 243=a_1(-27) \\\\ \dfrac{243}{-27}=a_1 \\\\ a_1=-9 .\end{array}
With $a_1=-9$ and $r=-3,$ then \begin{array}{l}\require{cancel} a_n=a_1r^{n-1} \\\\
a_n=-9(-3)^{n-1}
.\end{array}
With $n=5,$ the equation above bcomes \begin{array}{l}\require{cancel}
a_5=-9(-3)^{5-1}
\\\\
a_5=-9(-3)^{4}
\\\\
a_5=-9(81)
\\\\
a_5=-729
.\end{array}
Hence, $
a_5=-729 \text{ and } a_n=-9(-3)^{n-1}
.$