To find the five arithmetic means between 6 and -18, we need to insert five equally spaced numbers between them. This creates a sequence with seven terms: the first term is 6, the last term is -18, and the five arithmetic means are the intermediate terms.

### Step-by-Step Solution:

1. **Identify the Sequence:**
   - The sequence is: 6, \(a_2\), \(a_3\), \(a_4\), \(a_5\), \(a_6\), -18
   - Here, \(a_1 = 6\) and \(a_7 = -18\).

2. **Determine the Common Difference \(d\):**
   The common difference \(d\) can be found using the formula for the \(n\)th term of an arithmetic sequence:
   \[
   a_n = a_1 + (n-1) \times d
   \]
   For the 7th term (\(a_7\)):
   \[
   -18 = 6 + (7-1) \times d
   \]
   \[
   -18 = 6 + 6d
   \]
   \[
   6d = -18 - 6
   \]
   \[
   6d = -24
   \]
   \[
   d = \frac{-24}{6} = -4
   \]

3. **Find the Arithmetic Means:**
   Now that we have the common difference \(d = -4\), we can calculate the five arithmetic means:
   \[
   a_2 = a_1 + d = 6 - 4 = 2
   \]
   \[
   a_3 = a_2 + d = 2 - 4 = -2
   \]
   \[
   a_4 = a_3 + d = -2 - 4 = -6
   \]
   \[
   a_5 = a_4 + d = -6 - 4 = -10
   \]
   \[
   a_6 = a_5 + d = -10 - 4 = -14
   \]

### Final Answer:
The five arithmetic means between 6 and -18 are **2, -2, -6, -10, -14**.

Would you like more details or have any questions?

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Here are 5 related questions to further explore the concept:
1. How would you find the arithmetic means if there were only three means between two numbers?
2. What is the sum of the sequence 6, 2, -2, -6, -10, -14, -18?
3. How do you find the common difference if the number of arithmetic means is unknown?
4. Can you find the geometric means between 6 and -18?
5. What are the properties of an arithmetic sequence?

**Tip:** In an arithmetic sequence, the difference between any two consecutive terms is always the same, called the common difference.

Learn how to find five arithmetic means between 6 and -18 with a detailed step-by-step solution. This problem covers arithmetic sequences, common difference calculation, and term identification. Suitable for students in Grades 7-9.

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