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.