To insert 3 arithmetic means between 3 and -5, we need to determine four equal intervals (since we're inserting 3 means) between the first term (3) and the last term (-5). This forms an arithmetic sequence.

Step-by-Step Solution:

1. Identify the terms:

   • First term ($a_1$) = 3
   • Fifth term ($a_5$) = -5
   • Number of terms ($n$) = 5 (including the first term, the three means, and the last term)

2. Use the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$ where $d$ is the common difference.

3. Set up the equation using the 5th term: $a_5 = 3 + (5 - 1) \cdot d = -5$

4. Solve for the common difference ($d$): $-5 = 3 + 4d$ $4d = -8$ $d = -2$

5. Determine the arithmetic means:

   • Second term ($a_2$) = $a_1 + d = 3 - 2 = 1$
   • Third term ($a_3$) = $a_2 + d = 1 - 2 = -1$
   • Fourth term ($a_4$) = $a_3 + d = -1 - 2 = -3$

The sequence including the means:

• The sequence is: $3$, $1$, $-1$, $-3$, $-5$.

Therefore, the three arithmetic means are $1$, $-1$, and $-3$.

Would you like more details or have any questions?

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Here are 5 questions related to this topic:

1. How do you find the common difference in an arithmetic sequence?
2. What is the general formula for the nth term in an arithmetic sequence?
3. How do you calculate the sum of an arithmetic sequence?
4. What are the properties of an arithmetic sequence?
5. How does an arithmetic sequence differ from a geometric sequence?

Tip: Always double-check your common difference by substituting back into the sequence to ensure accuracy.