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This problem presents a sequence:

\[
z, \, 2, \, y, \, -3, \, x, \, -9, \, \dots
\]

We are told that this is an arithmetic sequence, which means each term is obtained by adding a constant common difference to the previous term. We need to find the value of \( x + y + z \).

### Step-by-Step Solution

1. **Identify the Common Difference**: 
   Since this is an arithmetic sequence, let's denote the common difference by \( d \).
   
2. **Set up Equations for Known Terms**:
   We have:
   - The second term is \( 2 \).
   - The fourth term is \( -3 \).
   - The sixth term is \( -9 \).

   These positions correspond to the following:
   
   - **First Term ( \( z \) )**
   - **Second Term**: \( z + d = 2 \)
   - **Fourth Term**: \( z + 3d = -3 \)
   - **Sixth Term**: \( z + 5d = -9 \)

3. **Form Equations**:
   From the above terms, we get the following equations:
   \[
   z + d = 2
   \]
   \[
   z + 3d = -3
   \]
   \[
   z + 5d = -9
   \]

4. **Solve the System of Equations**:
   - From the first equation \( z + d = 2 \), we can solve for \( z \) as \( z = 2 - d \).
   - Substitute \( z = 2 - d \) into the second equation:
     \[
     (2 - d) + 3d = -3
     \]
     Simplifying, we get:
     \[
     2 + 2d = -3 \Rightarrow 2d = -5 \Rightarrow d = -\frac{5}{2}
     \]
   - Substitute \( d = -\frac{5}{2} \) back into \( z = 2 - d \):
     \[
     z = 2 - \left(-\frac{5}{2}\right) = 2 + \frac{5}{2} = \frac{9}{2}
     \]

5. **Calculate \( y \) and \( x \)**:
   - \( y \) is the third term, so:
     \[
     y = z + 2d = \frac{9}{2} + 2 \times \left(-\frac{5}{2}\right) = \frac{9}{2} - 5 = \frac{-1}{2}
     \]
   - \( x \) is the fifth term, so:
     \[
     x = z + 4d = \frac{9}{2} + 4 \times \left(-\frac{5}{2}\right) = \frac{9}{2} - 10 = -\frac{11}{2}
     \]

6. **Calculate \( x + y + z \)**:
   \[
   x + y + z = -\frac{11}{2} + \frac{-1}{2} + \frac{9}{2} = \frac{-11 - 1 + 9}{2} = \frac{-3}{2}
   \]

### Final Answer
\[
x + y + z = -\frac{3}{2}
\]

Would you like a further breakdown, or do you have any questions?

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### Related Questions
1. What is an arithmetic sequence, and how do we identify its properties?
2. How do you solve a system of equations involving sequences?
3. What is the significance of a common difference in an arithmetic sequence?
4. How does changing the first term affect the entire arithmetic sequence?
5. What are the conditions for a sequence to be classified as arithmetic?

### Tip
When solving sequence problems, always check the terms carefully and use known positions to set up equations for the common difference and first term.

اگر ...،-۳،x،۷،y،-۹،z،... یک دنباله حسابی باشد، حاصل x + y + z کدام است؟

Step-by-step solution to find the sum x + y + z in an arithmetic sequence where terms include -3, x, 7, y, -9, and z. Learn how to identify the common difference and solve using a system of equations. Suitable for Grades 9-12.

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