<Question>
Consider the geometric sequence $\frac{125}{9}, \frac{25}{3}, 5, 3, \ldots$. What is the eighth term of the sequence? Express your answer as a common fraction.
</Question>

<think>
<Plan_1>
Okay, so I need to find the eighth term of the geometric sequence given: 125/9, 25/3, 5, 3, .... Let me start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as r. So, to find the eighth term, I probably need to figure out the common ratio first.

<prompt_1.1> Identify the first term and calculate the common ratio of the given geometric sequence. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me check the given terms to confirm the common ratio. The first term is 125/9. The second term is 25/3. To find the common ratio, I can divide the second term by the first term. Let me compute that: (25/3) divided by (125/9). Dividing fractions is the same as multiplying by the reciprocal, so (25/3) * (9/125). Let's calculate that: 25 and 125 have a common factor of 25, so 25/125 is 1/5. Similarly, 9 and 3 have a common factor of 3, so 9/3 is 3. So multiplying 1/5 and 3 gives 3/5. Therefore, the common ratio r is 3/5.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, let me verify that with the next terms. If the second term is 25/3, multiplying by 3/5 should give the third term. Let's check: (25/3) * (3/5) = (25*3)/(3*5) = 75/15 = 5.
That's correct, the third term is 5. Then, multiplying 5 by 3/5 gives 15/5 = 3, which is the fourth term. So the common ratio r is indeed 3/5.

Now, the general formula for the nth term of a geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term. Here, we need the eighth term, so n = 8. Let's plug in the values.

<prompt_2.1> Use the general formula for the nth term of a geometric sequence to find the eighth term. </prompt_2.1>
-----

But let me check with another approach. Maybe writing out the terms step by step. Let's list the terms:

Term 1: 125/9

<prompt_2.2> List the terms of the sequence step by step to verify the eighth term. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
The first term a_1 is 125/9. The common ratio r is 3/5. So, a_8 = (125/9) * (3/5)^(8-1) = (125/9) * (3/5)^7.

Wait, let me compute (3/5)^7 first. But maybe there's a way to simplify this before multiplying everything out. Let's see.

First, note that 125 is 5^3, and 9 is 3^2. Let's write all the components in terms of their prime factors to see if things cancel out. 

125/9 is (5^3)/(3^2). Then, (3/5)^7 is (3^7)/(5^7). Multiplying these together: (5^3 / 3^2) * (3^7 / 5^7) = (5^3 * 3^7) / (3^2 * 5^7). Simplify the exponents:

For the numerator: 5^3 * 3^7

Denominator: 3^2 * 5^7

So, we can subtract exponents for like bases. For 5: 3 - 7 = -4, so 5^(-4) which is 1/5^4. For 3: 7 - 2 = 5, so 3^5. Therefore, the expression simplifies to (3^5)/(5^4).
</execution_2.1>

<execution_2.2>
Term 2: (125/9) * (3/5) = (125*3)/(9*5) = 375/45 = 25/3 ✔️

Term 3: (25/3) * (3/5) = (25*3)/(3*5) = 75/15 = 5 ✔️

Term 4: 5 * (3/5) = 15/5 = 3 ✔️

Term 5: 3 * (3/5) = 9/5

Term 6: (9/5) * (3/5) = 27/25

Term 7: (27/25) * (3/5) = 81/125

Term 8: (81/125) * (3/5) = 243/625 ✔️
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that again.

<prompt_3.1> Verify the simplification of the expression for the eighth term. </prompt_3.1>

Based on execution_2.1, execution_2.2:
Hmm, so that seems to check out. Let me confirm once more.

Alternatively, let's compute (3/5)^7. Let's see:

<prompt_3.2> Re-evaluate the expression for the eighth term and confirm the result. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Wait, (5^3 * 3^7) / (3^2 * 5^7) = 3^(7-2) * 5^(3-7) = 3^5 * 5^(-4) = 3^5 / 5^4. Yes, that's correct. 3^5 is 243 and 5^4 is 625. So, 243/625.
</execution_3.1>

<execution_3.2>
(3/5)^1 = 3/5

(3/5)^2 = 9/25

(3/5)^3 = 27/125

(3/5)^4 = 81/625

(3/5)^5 = 243/3125

(3/5)^6 = 729/15625

(3/5)^7 = 2187/78125

Then, multiplying this by 125/9:

(125/9) * (2187/78125)

Let's compute this. First, simplify the fractions. 125 and 78125: 78125 divided by 125 is 625, because 125*625 = 78125. So 125/78125 = 1/625.

Similarly, 2187 and 9: 2187 divided by 9 is 243, since 9*243 = 2187. So 2187/9 = 243.

Therefore, (125/9) * (2187/78125) = (1/625) * 243 = 243/625.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_3.2, execution_2.2:
So both methods give the same answer, which is reassuring. Therefore, the eighth term is 243/625.

<prompt_4.1> Compare results from both methods to confirm the eighth term. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Let me just make sure that I didn't make a mistake in the exponents. Let's see:

Starting with a_8 = (125/9) * (3/5)^7.

125 is 5^3, so 5^3 * (3^7) / (5^7) = 3^7 / 5^4. Then, 3^7 is 2187 and 5^4 is 625. Then, divided by 9? Wait, no. Wait, the initial term is 125/9, so it's (5^3 / 3^2) * (3^7 / 5^7) = (5^3 / 5^7) * (3^7 / 3^2) = (1/5^4) * (3^5) = 3^5 / 5^4 = 243 / 625. Yes, that's correct. Because 3^5 is 243 and 5^4 is 625. So 243/625 is the simplified fraction. 

Yes, that's correct. So I think that's the right answer. Just to recap, since each term is multiplied by 3/5, starting from 125/9, after seven multiplications by 3/5, we get to the eighth term, which is 125/9 * (3/5)^7 = 243/625.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the eighth term of the geometric sequence is $\boxed{\frac{243}{625}}$.
</Final_answer>
</think>