EXPLANATION:
We are given a sequence and the first and second terms are;
[tex]\begin{gathered} a_1=-2 \\ a_2=\frac{2}{3} \end{gathered}[/tex]The common ratio is determined by dividing a term by its preceeding term. Hence, we can derive the common ratio by dividing the second term by the first.
[tex]\begin{gathered} Common\text{ }ratio: \\ r=\frac{2}{3}\div\frac{-2}{1} \end{gathered}[/tex][tex]r=\frac{2}{3}\times\frac{1}{-2}[/tex][tex]r=-\frac{1}{3}[/tex]The recursive formula for this sequence would be given as follows;
[tex]a_n=a_{n-1}\times r[/tex]Where the variables are;
[tex]\begin{gathered} a_n=nth\text{ term} \\ r=-\frac{1}{3} \end{gathered}[/tex]We now have;
[tex]a_n=a_{n-1}(-\frac{1}{3})[/tex]For the 5th term we would have;
[tex]a_5=a_4(-\frac{1}{3})[/tex][tex]a_5=\frac{2}{27}(-\frac{1}{3})[/tex][tex]a_5=-\frac{2}{81}[/tex]For the 6th term we would have;
[tex]a_6=a_5(-\frac{1}{3})[/tex][tex]a_6=-\frac{2}{81}\times(-\frac{1}{3})[/tex][tex]a_6=\frac{2}{243}[/tex]For the 7th term we would have;
[tex]a_7=a_6(-\frac{1}{3})[/tex][tex]a_7=\frac{2}{243}(-\frac{1}{3})[/tex][tex]a_7=-\frac{2}{729}[/tex]ANSWER:
[tex]\begin{gathered} Recursive\text{ }formula: \\ a_n=a_{n-1}(-\frac{1}{3}) \\ 7th\text{ }term: \\ a_7=-\frac{2}{729} \end{gathered}[/tex]