Solution:
Given the sequence;
[tex]-3,6,-12,24[/tex]
The common ratio is the ratio between two consecutive numbers in a geometric sequence.
Thus;
[tex]\begin{gathered} r=\frac{a_2}{a_1}=\frac{a_3}{a_2} \\ \\ r=\frac{6}{-3} \\ \\ r=-2 \end{gathered}[/tex]
Common Ratio:
[tex]r=-2[/tex]
Also, given the formula;
[tex]a_n=-3\cdot(-2)^{n-1}[/tex]
The formula is an explicit formula of the geometric sequence.
The recursive formula is;
[tex]\begin{gathered} a_n=r(a_{n-1}) \\ \\ a_n=-2(a_{n-1}) \end{gathered}[/tex]
Then, the ninth term is;
[tex]\begin{gathered} n=9 \\ \\ a_9=-3\cdot(-2)^{9-1} \\ \\ a_9=-3\cdot(-2)^8 \\ \\ a_9=-3(256) \\ \\ a_9=-768 \end{gathered}[/tex]
The ninth term is;
[tex]a_9=-768[/tex]
