To find two arithmetic means between $-5$ and $-17$, we need to determine the three terms of the arithmetic sequence where the first term ($a_1$) is $-5$, the last term ($a_n$) is $-17$, and there are four terms in total.

The general form of an arithmetic sequence is given by: $a_n = a_1 + (n - 1)d$

Where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $n$ is the number of terms
• $d$ is the common difference

Since we need to insert two arithmetic means between $-5$ and $-17$, the sequence will have 4 terms in total: $a_1, a_2, a_3, a_4$ with:

• $a_1 = -5$
• $a_4 = -17$
• $n = 4$

We can plug these values into the formula for the nth term: $a_4 = a_1 + (4 - 1)d$ $-17 = -5 + 3d$

Solving for $d$: $-17 = -5 + 3d$ $-17 + 5 = 3d$ $-12 = 3d$ $d = \frac{-12}{3}$ $d = -4$

Now that we have the common difference $d = -4$, we can find the intermediate terms: $a_2 = a_1 + d = -5 + (-4) = -9$ $a_3 = a_1 + 2d = -5 + 2(-4) = -5 - 8 = -13$

Thus, the two arithmetic means between $-5$ and $-17$ are: $-9 \text{ and } -13$

So, the sequence is: $-5, -9, -13, -17$

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you determine the common difference in an arithmetic sequence?
2. What is the formula for finding the nth term of an arithmetic sequence?
3. How do you insert multiple arithmetic means between two given numbers?
4. Can you provide an example with a different set of numbers?
5. What happens to the common difference if the terms in the sequence are increased?
6. How can you verify if a sequence is arithmetic?
7. What is the sum of the terms in the sequence $-5, -9, -13, -17$?
8. How do arithmetic sequences apply in real-life scenarios?

Tip: When solving problems involving arithmetic sequences, always write down the known values and the formulas you need to use. This will help you organize your work and ensure you don't miss any steps.