To solve the problem, we must first identify the terms immediately before $x$ and after $2y - x$ in the arithmetic sequence.

Step 1: Recall the property of an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. If the four terms $x, \frac{x}{2}, y-2, 2y-x$ form an arithmetic sequence, then: $\text{Common difference} = \text{(Term 2 - Term 1)} = \text{(Term 3 - Term 2)} = \text{(Term 4 - Term 3)}.$

Step 2: Set up equations

1. From the first two terms: $\frac{x}{2} - x = -\frac{x}{2}.$ The common difference is $-\frac{x}{2}$.

2. From the second and third terms: $(y - 2) - \frac{x}{2} = -\frac{x}{2}.$ Simplify: $y - 2 - \frac{x}{2} = -\frac{x}{2}.$ Rearrange: $y - 2 = 0 \quad \Rightarrow \quad y = 2.$

3. From the third and fourth terms: $(2y - x) - (y - 2) = -\frac{x}{2}.$ Substitute $y = 2$: $(2(2) - x) - (2 - 2) = -\frac{x}{2}.$ Simplify: $4 - x = -\frac{x}{2}.$ Multiply through by 2 to eliminate the fraction: $8 - 2x = -x \quad \Rightarrow \quad 8 = x.$

Step 3: Verify the terms

• First term: $x = 8$,
• Second term: $\frac{x}{2} = 4$,
• Third term: $y - 2 = 2 - 2 = 0$,
• Fourth term: $2y - x = 4 - 8 = -4$.

These form an arithmetic sequence with a common difference of $-4$: $8, 4, 0, -4.$

Step 4: Find the terms immediately before $x$ and after $2y - x$

1. The term before $x = 8$ is: $8 + (-4) = 12.$

2. The term after $2y - x = -4$ is: $-4 + (-4) = -8.$

Final Answer:

The terms immediately before $x$ and after $2y - x$ are: $\boxed{12 \text{ and } -8.}$

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Related Questions:

1. What are the general properties of an arithmetic sequence?
2. How do you determine the $n$-th term of an arithmetic sequence?
3. Can you generalize this solution for different starting terms?
4. How does this sequence change if the common difference is positive instead of negative?
5. How are arithmetic and geometric sequences different?

Tip:

Always verify your solution by substituting the values back into the sequence to ensure they form a valid arithmetic progression.