Let's analyze the given sequence to identify the pattern and find the 9th term.

The first three terms of the sequence are:

1. $a_1 = 3$
2. $a_2 = 5$
3. $a_3 = \frac{25}{3}$

To find the pattern, let's check if there's a consistent way the terms are progressing. A common approach is to check if the sequence might be arithmetic or geometric.

Checking if the sequence is arithmetic:

An arithmetic sequence has a constant difference between consecutive terms. Let's check the differences:

1. Difference between the 2nd and 1st term: $5 - 3 = 2$

2. Difference between the 3rd and 2nd term: $\frac{25}{3} - 5 = \frac{25}{3} - \frac{15}{3} = \frac{10}{3} \approx 3.333$

Since the differences are not equal, the sequence is not arithmetic.

Checking if the sequence is geometric:

A geometric sequence has a constant ratio between consecutive terms. Let's check the ratios:

1. Ratio between the 2nd and 1st term: $\frac{5}{3} \approx 1.6667$

2. Ratio between the 3rd and 2nd term: $\frac{\frac{25}{3}}{5} = \frac{25}{3} \times \frac{1}{5} = \frac{25}{15} = \frac{5}{3} \approx 1.6667$

Since the ratios are equal, the sequence is geometric with a common ratio $r = \frac{5}{3}$.

Finding the 9th term:

The formula for the $n$th term of a geometric sequence is: $a_n = a_1 \times r^{n-1}$

For $n = 9$: $a_9 = 3 \times \left(\frac{5}{3}\right)^{8}$

Let's calculate $a_9$:

$a_9 = 3 \times \left(\frac{5}{3}\right)^{8} = 3 \times \left(\frac{5^8}{3^8}\right)$

$a_9 = \frac{3 \times 5^8}{3^8} = \frac{5^8}{3^7}$

Now, compute $5^8$ and $3^7$:

$5^8 = 390625 \quad \text{and} \quad 3^7 = 2187$

$a_9 = \frac{390625}{2187} \approx 178.610$

Rounding to the nearest thousandth:

$a_9 \approx 178.610$

Thus, the 9th term of the sequence is approximately 178.610.

Do you want further details or have any questions? Here are five related questions that might interest you:

1. How do you determine whether a sequence is arithmetic or geometric?
2. What is the significance of the common ratio in a geometric sequence?
3. How do you calculate the sum of a geometric sequence?
4. Can a sequence have both arithmetic and geometric properties?
5. How does the choice of the common ratio affect the growth of a geometric sequence?

Tip: When dealing with sequences, identifying the pattern early on (arithmetic, geometric, etc.) makes it easier to predict future terms.