find the three-arithmetic means between 1 to 17

To find the three arithmetic means between 1 and 17, we can follow these steps:

### Step 1: General formula for arithmetic means
If we want to insert three arithmetic means between two numbers \(a_1\) (the first term) and \(a_5\) (the last term), we need to find four equal differences \(d\) between consecutive terms. The sequence will look like this:

\[
a_1, a_2, a_3, a_4, a_5
\]

Where:
- \(a_1 = 1\)
- \(a_5 = 17\)
- \(a_2, a_3, a_4\) are the arithmetic means we want to find.

### Step 2: Formula for the nth term in an arithmetic sequence
The general formula for an arithmetic sequence is:

\[
a_n = a_1 + (n - 1)d
\]

For \(n = 5\), we know:

\[
a_5 = a_1 + 4d
\]

### Step 3: Calculate the common difference \(d\)
We substitute the known values:

\[
17 = 1 + 4d
\]

Now solve for \(d\):

\[
17 - 1 = 4d
\]
\[
16 = 4d
\]
\[
d = 4
\]

### Step 4: Find the arithmetic means
Now that we have \(d = 4\), we can calculate the values of \(a_2\), \(a_3\), and \(a_4\) using the formula for the nth term:

- \(a_2 = a_1 + d = 1 + 4 = 5\)
- \(a_3 = a_1 + 2d = 1 + 2(4) = 9\)
- \(a_4 = a_1 + 3d = 1 + 3(4) = 13\)

### Conclusion:
The three arithmetic means between 1 and 17 are:

\[
5, 9, 13
\]

Would you like a detailed explanation of any step? 

Here are some related questions:
1. How do you find the arithmetic mean between two numbers?
2. What is the formula for the sum of an arithmetic sequence?
3. How does the concept of arithmetic means differ from geometric means?
4. How do you find more than three arithmetic means between two numbers?
5. What are common applications of arithmetic sequences?

**Tip:** Always verify your common difference \(d\) by checking the final term of the sequence matches the expected value.

Learn how to find the three arithmetic means between 1 and 17. This step-by-step solution covers key concepts in arithmetic sequences and shows how to use the formula for the nth term. Perfect for students in Grades 6-8 looking to enhance their understanding of sequences.

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