The given sequence is:

$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \dots$

This is a geometric sequence where the first term $a = \frac{2}{3}$ and the common ratio $r$ is:

$r = \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{4}{9} \times \frac{3}{2} = \frac{2}{3}$

The general formula for the $n$-th term of a geometric sequence is:

$a_n = a \cdot r^{n-1}$

Substituting the known values to find the 10th term:

$a_{10} = \frac{2}{3} \cdot \left( \frac{2}{3} \right)^{10-1} = \frac{2}{3} \cdot \left( \frac{2}{3} \right)^9$

Now let's compute:

$a_{10} = \frac{2}{3} \cdot \frac{2^9}{3^9} = \frac{2^{10}}{3^{10}}$

$a_{10} = \frac{1024}{59049}$

Thus, the 10th term of the sequence is:

$\frac{1024}{59049}$

Would you like more details on any part of this solution?

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Here are 5 related questions:

1. How do you find the sum of the first $n$ terms of a geometric sequence?
2. What is the common ratio in a geometric sequence?
3. How do you determine if a sequence is arithmetic or geometric?
4. Can geometric sequences have negative common ratios?
5. How do you find the limit of a geometric sequence as $n \to \infty$?

Tip: Always check for the common ratio by dividing consecutive terms when determining if a sequence is geometric.