An **arithmetic sequence** (or arithmetic progression) is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the **common difference**.

### Key Characteristics of an Arithmetic Sequence:

1. **Common Difference (d):** The difference between consecutive terms is always the same. If the first term is \(a_1\) and the second term is \(a_2\), the common difference \(d\) is given by:
   \[
   d = a_2 - a_1
   \]

2. **General Form:** The \(n\)-th term (\(a_n\)) of an arithmetic sequence can be found using the formula:
   \[
   a_n = a_1 + (n-1) \cdot d
   \]
   where:
   - \(a_1\) is the first term,
   - \(d\) is the common difference,
   - \(n\) is the term number.

3. **Example:** Consider the sequence: \(2, 5, 8, 11, 14, \ldots\)
   - Here, the common difference \(d = 5 - 2 = 3\).
   - The sequence follows the pattern where each term is 3 more than the previous one.

4. **Sum of an Arithmetic Sequence:** The sum of the first \(n\) terms (\(S_n\)) of an arithmetic sequence can be calculated using the formula:
   \[
   S_n = \frac{n}{2} (a_1 + a_n)
   \]
   or equivalently,
   \[
   S_n = \frac{n}{2} (2a_1 + (n-1)d)
   \]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the formula for the sum of the first \(n\) terms in an arithmetic sequence?
2. How do you derive the general formula for the \(n\)-th term of an arithmetic sequence?
3. What is the difference between an arithmetic sequence and a geometric sequence?
4. How can you determine if a given sequence is arithmetic?
5. Can an arithmetic sequence have a common difference of 0?

**Tip:** Understanding arithmetic sequences is essential for solving problems involving patterns, series, and linear equations.

Learn about arithmetic sequences, including how to identify the common difference, calculate the n-th term, and find the sum of terms. This guide covers essential algebraic formulas and is suitable for students in Grades 7-10.

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