ANSWER
[tex] \boxed {a_n= - \frac{1}{9} {( - 3)}^{n - 1} }[/tex]
EXPLANATION
The given geometric sequence is
[tex] - \frac{1}{9} , \frac{1}{3} , - 1,3, - 9,...[/tex]
The first term of the geometric sequence is,
[tex]a_1= - \frac{1}{9} [/tex]
and there is a common ratio of
[tex]r = \frac{ \frac{1}{3} }{ - \frac{1}{9} } = - 3[/tex]
The explicit formula, for the geometric sequence is given by,
[tex]a_n=a_1 {r}^{n - 1} [/tex]
We substitute the values into the formula to get the explicit formula as,
[tex]a_n= - \frac{1}{9} {( - 3)}^{n - 1} [/tex]
