Answer:
1280/2187
Step-by-step explanation:
We usually study arithmetic and geometric sequences. The terms of an arithmetic sequence have a common difference. The terms of a geometric sequence have a common ratio. You can tell what kind of a sequence it is by determining if the difference is constant of the ratio is constant.
__
Here, the difference of the first two terms is 10 -15 = -5. The next term of an arithmetic sequence would be 10 +(-5) = 5. The next term is not that, but is 20/3.
The ratio of the first two terms is 10/15 = 2/3. If the sequence is geometric, the next term will be 10(2/3) = 20/3, which it is. This geometric sequence has first term 15 and common ratio 2/3.
__
The general term of a geometric sequence is ...
[tex]a_n=a_1\cdot r^{n-1}\qquad\text{$n^{th}$ term with first term $a_1$ and common ratio $r$}[/tex]
You want the 9th term of the given sequence. It is ...
[tex]a_9=15\cdot\left(\dfrac{2}{3}\right)^{9-1}=5\cdot 2^8\cdot3^{-7}\\\\\boxed{a_9=\dfrac{1280}{2187}}[/tex]
