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?

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.