a10:{1000,100,10,...} solve this geometric sequence

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 ( \( |r| < 1 \) ).

Discover how to solve for the 10th term of the geometric sequence {1000, 100, 10, ...}. This step-by-step solution explains key concepts in sequences and series, using the geometric sequence formula to calculate the 10th term as 0.000001. Ideal for students in Grades 9-12.

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