An arithmetic sequence has a constant difference between consecutive terms. Given that the first three terms are $k - 2$, $k + 2$, and $k$, we can use the property of arithmetic sequences that the difference between consecutive terms is constant.

The difference between the second and first terms must equal the difference between the third and second terms:

$(k + 2) - (k - 2) = k - (k + 2)$

Step-by-step solution:

1. Simplify the left-hand side: $(k + 2) - (k - 2) = k + 2 - k + 2 = 4$

2. Simplify the right-hand side: $k - (k + 2) = k - k - 2 = -2$

Now, we set up the equation: $4 = -2$ This is a contradiction, meaning there is no value of $k$ that satisfies these conditions. Therefore, the given terms $k - 2$, $k + 2$, and $k$ cannot form an arithmetic sequence for any value of $k$.

Conclusion:

There are no possible values of $k$ that make $k - 2$, $k + 2$, and $k$ the first three terms of an arithmetic sequence.

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Related questions:

1. What is an arithmetic sequence, and how can you determine the common difference?
2. How would the equation change if the first three terms were $k-1$, $k+1$, and $k+3$?
3. What is a geometric sequence, and how is it different from an arithmetic sequence?
4. How do you find the $n$-th term of an arithmetic sequence?
5. Can you solve arithmetic progression problems using a system of equations?

Tip: Always verify if terms satisfy sequence properties by testing with real values for insight into the sequence behavior.