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.