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The problem provides the first three terms of a sequence: \(40\), \(24\), and \(\frac{72}{5}\), and asks us to find the 10th term. Let's determine the type of sequence and find the 10th term.

### Step 1: Identify the Pattern
To identify if this is an arithmetic or geometric sequence, we calculate the differences and ratios between terms:

1. **Difference (for arithmetic sequences):**
   \[
   24 - 40 = -16
   \]
   \[
   \frac{72}{5} - 24 = \frac{72}{5} - \frac{120}{5} = -\frac{48}{5} \approx -9.6
   \]
   The differences aren't consistent, so it's not an arithmetic sequence.

2. **Ratio (for geometric sequences):**
   \[
   \frac{24}{40} = 0.6
   \]
   \[
   \frac{\frac{72}{5}}{24} = \frac{72}{5} \cdot \frac{1}{24} = \frac{72}{120} = 0.6
   \]
   The ratios are consistent, indicating this is a **geometric sequence** with a common ratio of \(0.6\).

### Step 2: Use the Geometric Sequence Formula
For a geometric sequence, the \(n\)-th term \(a_n\) is given by:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where:
- \(a_1 = 40\) (the first term),
- \(r = 0.6\) (the common ratio),
- \(n = 10\).

### Step 3: Calculate the 10th Term
\[
a_{10} = 40 \cdot (0.6)^{9}
\]

Calculating \( (0.6)^9 \) and multiplying by 40:
\[
a_{10} \approx 40 \cdot 0.010077696 = 0.4031
\]

### Final Answer
The 10th term, rounded to the nearest thousandth, is approximately:
\[
\boxed{0.403}
\]

Would you like further clarification on any part? 

### Related Questions
1. How do you find the 5th term in a geometric sequence with a different common ratio?
2. What happens if a geometric sequence has a ratio greater than 1?
3. How do you determine the sum of the first 10 terms in a geometric sequence?
4. Can an arithmetic sequence have a fractional difference?
5. How can you verify if a sequence is neither arithmetic nor geometric?

### Tip
For a geometric sequence, always verify the ratio between consecutive terms remains constant.

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). Find the 10th term given the sequence: 40, 24, 72/5.

Learn how to determine the 10th term in a geometric sequence with initial terms 40, 24, and 72/5. This problem covers the formula for geometric sequences and finding terms using common ratios. Suitable for students in Grades 9-12.

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