The sequence provided is \(\{-5, -2, 1, 4, 7\}\). Let's analyze it to identify a pattern or relationship.

First, let's look at the differences between consecutive terms:

\[
\begin{aligned}
&-2 - (-5) = 3,\\
&1 - (-2) = 3,\\
&4 - 1 = 3,\\
&7 - 4 = 3.
\end{aligned}
\]

The difference between each consecutive term is constant and equal to 3. This indicates that the sequence is an arithmetic sequence where each term is obtained by adding a common difference (\(d = 3\)) to the previous term.

The general form for an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]

Where:
- \(a_n\) is the \(n\)-th term.
- \(a_1\) is the first term.
- \(d\) is the common difference.
- \(n\) is the term number.

For this sequence:
- \(a_1 = -5\)
- \(d = 3\)

Thus, the \(n\)-th term can be calculated as:
\[
a_n = -5 + (n-1) \times 3
\]

Let's verify this formula with the given terms:
- For \(n = 1\): \(a_1 = -5 + (1-1) \times 3 = -5\)
- For \(n = 2\): \(a_2 = -5 + (2-1) \times 3 = -2\)
- For \(n = 3\): \(a_3 = -5 + (3-1) \times 3 = 1\)
- For \(n = 4\): \(a_4 = -5 + (4-1) \times 3 = 4\)
- For \(n = 5\): \(a_5 = -5 + (5-1) \times 3 = 7\)

All terms match the provided sequence. Thus, the formula for the \(n\)-th term is correct.

If you have any further questions or need more details, please let me know.

Here are 8 related questions to expand your understanding:

1. What is the definition of an arithmetic sequence?
2. How do you derive the formula for the \(n\)-th term of an arithmetic sequence?
3. Can you find the sum of the first \(n\) terms of an arithmetic sequence?
4. What is the relationship between arithmetic sequences and linear functions?
5. How do you determine the common difference in an arithmetic sequence given two terms?
6. Can you solve for a specific term in an arithmetic sequence if you know its position?
7. How do arithmetic sequences appear in real-world problems or applications?
8. What are the differences between arithmetic and geometric sequences?

**Tip:** When analyzing sequences, always look for patterns in the differences between terms. If the differences are constant, the sequence is likely arithmetic. If the ratios between terms are constant, the sequence is likely geometric.

Explore the arithmetic sequence {-5, -2, 1, 4, 7} with a detailed analysis. Learn how to identify patterns, calculate differences between terms, and derive the formula for the nth term of an arithmetic sequence. Suitable for students in Grades 9-12.

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