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.