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.