The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).
3, comma, 5, comma, start fraction, 25, divided by, 3, end fraction, comma, point, point, .
3,5, 
3
25
​
 ,...
Find the 9th term.


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.

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). 3, 5, 25/3, ... Find the 9th term.

Learn how to find the 9th term of a geometric sequence where the first three terms are 3, 5, and 25/3. This solution involves identifying the common ratio and using the formula for the nth term of a geometric sequence. Suitable for students in Grades 9-12.

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