What are the first and last terms of an arithmetic sequence when its arithmetic means are 8 and 13?

In an arithmetic sequence, the arithmetic mean between two terms \(a_n\) and \(a_{n+2}\) is equal to the term \(a_{n+1}\) that lies between them. The arithmetic means 8 and 13 in your question indicate the values of some middle terms in the sequence.

To solve for the first and last terms, let's assume the arithmetic sequence is given by:

\[ a_1, a_2, a_3, \ldots, a_{n-1}, a_n \]

### Step 1: Establish relationships using the given means.
- Let the first arithmetic mean (8) be \(a_{n}\), so:
  \[
  a_n = 8
  \]
- Let the second arithmetic mean (13) be \(a_{n+1}\), so:
  \[
  a_{n+1} = 13
  \]

### Step 2: Calculate the common difference.
The difference between consecutive terms is constant in an arithmetic sequence. The common difference \(d\) can be calculated using the formula:

\[
d = a_{n+1} - a_n = 13 - 8 = 5
\]

### Step 3: Express the first term \(a_1\).
Since each term in the sequence is related to the first term by the common difference, we can write:

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

For \(a_n = 8\), and the sequence number of \(a_n\) is unknown, but \(a_1\) can be solved:

### Step 4: Express the last term \(a_m\).
The last term \(a_m\) would be calculated using the common difference:

\[
a_m = a_1 + (m-1)d
\]

### Example solution:
Assume \(a_3 = 8\) and \(a_4 = 13\):
- Then \(a_1 = 8 - 2d = 8 - 10 = -2\)
- And \(a_m = 13 + 5 = 18\)

### Final answer:
- The first term \(a_1 = -2\)
- The last term \(a_m = 18\)

Would you like details on how to generalize or solve other questions related to this?

### Relative Questions:
1. What is the sum of the first 20 terms of the arithmetic sequence?
2. How can we determine the total number of terms in the sequence?
3. What happens to the sequence if the common difference is doubled?
4. How do you find the arithmetic mean of the entire sequence?
5. What are the conditions for a sequence to be considered arithmetic?
6. How does changing the first term affect the entire sequence?
7. Can you provide examples where the arithmetic means are negative?
8. How do arithmetic sequences apply to real-world problems?

**Tip:** Understanding how the common difference relates to the terms is key to mastering arithmetic sequences.

Explore how to find the first and last terms of an arithmetic sequence when given arithmetic means of 8 and 13. This problem involves understanding arithmetic sequences, calculating common differences, and determining sequence endpoints. Suitable for Grades 9-12.

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