The sequence provided is a geometric sequence: $a_{10} : \{ 1000, 100, 10, \dots \}$.

Step-by-step Solution:

1. Identify the first term ($a_1$): The first term of the sequence is 1000.

2. Identify the common ratio ($r$): To find the common ratio, divide any term by the previous term. Let's divide the second term by the first term: $r = \frac{100}{1000} = 0.1$

   So, the common ratio is $r = 0.1$.

3. General formula for the $n$-th term of a geometric sequence: The formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Substituting $a_1 = 1000$ and $r = 0.1$, we get: $a_n = 1000 \cdot (0.1)^{n-1}$

4. Find the 10th term ($a_{10}$): Using the formula above for $n = 10$: $a_{10} = 1000 \cdot (0.1)^{10-1} = 1000 \cdot (0.1)^9$ $a_{10} = 1000 \cdot 0.000000001 = 0.000001$

Thus, the 10th term of the sequence is $a_{10} = 0.000001$.

Would you like further details or have any questions?

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Here are 5 related questions:

1. How do you calculate the sum of the first 10 terms of a geometric sequence?
2. What happens if the common ratio is negative in a geometric sequence?
3. How would you find the sum of an infinite geometric series?
4. What are the conditions for a geometric series to converge?
5. How does the common ratio affect the growth or decay of the sequence?

Tip: Always check the value of the common ratio to determine if the sequence is growing ( $|r| > 1$ ) or decaying ( $|r| < 1$ ).