https://file.aiquickdraw.com/mathbot/file/1730608907221us55whuk.jpg

To solve this problem, let's examine the given sequence:

The sequence is defined as:
\[
\{ a_1, a_2, a_3, a_4, \dots \} = \left\{ \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \dots \right\}
\]

Notice that each term is a fraction with the numerator \( 3 \) and a denominator that is a power of \( 2 \). Let's write out the terms to find a pattern:

1. \( a_1 = \frac{3}{2} = \frac{3}{2^1} \)
2. \( a_2 = \frac{3}{4} = \frac{3}{2^2} \)
3. \( a_3 = \frac{3}{8} = \frac{3}{2^3} \)
4. \( a_4 = \frac{3}{16} = \frac{3}{2^4} \)

It appears that each term \( a_n \) can be written as:
\[
a_n = \frac{3}{2^n}
\]

We are asked to determine \( a_9 \).

### Calculation of \( a_9 \)
Using the formula we identified:
\[
a_9 = \frac{3}{2^9}
\]

Calculating \( 2^9 \):
\[
2^9 = 512
\]

Thus:
\[
a_9 = \frac{3}{512}
\]

### Final Answer
\[
a_9 = \frac{3}{512}
\]

Would you like further details, or do you have any questions?

Here are some related questions to expand your understanding:
1. How would you find the general term \( a_n \) for any other sequence?
2. What is the sum of the first \( n \) terms of this sequence?
3. How can we determine if this sequence converges as \( n \to \infty \)?
4. What would the value of \( a_{10} \) be in this sequence?
5. How would this sequence look if the numerator was a different constant, like \( 5 \) instead of \( 3 \)?

**Tip:** For geometric sequences, identifying the common ratio between terms can simplify finding any term in the sequence.

Given the sequence {a1, a2, a3, a4,...} = {3/2, 3/4, 3/8, 3/16,...}, determine a9.

Learn how to find the 9th term (a9) in a geometric sequence defined as a_n = 3 / 2^n. This problem covers sequence concepts and formulas, focusing on geometric progression. Suitable for students in Grades 9-10.

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