In an arithmetic sequence, each term after the first is obtained by adding a constant difference (denoted by $d$) to the previous term. The general form for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) d$

To express a sum of terms from the first to the $n$-th term in summation notation, we write:

$S_n = \sum_{k=1}^{n} a_1 + (k - 1) d$

Here:

• $a_1$ is the first term of the sequence.
• $d$ is the common difference between consecutive terms.
• The summation runs from $k = 1$ to $k = n$, where each term is represented by $a_1 + (k - 1) d$.

Example:

If the sequence is $2, 5, 8, 11, \dots$, with the first term $a_1 = 2$ and the common difference $d = 3$, the sum of the first $n$ terms can be written as:

$S_n = \sum_{k=1}^{n} 2 + (k - 1) 3$

This represents the sum of the first $n$ terms of the arithmetic sequence.

---

Would you like more details on this? Or perhaps an example to work through?

Here are some related questions you might find interesting:

1. How do you find the sum of an arithmetic sequence using a formula?
2. What is the formula for the $n$-th term in an arithmetic sequence?
3. How do you identify the common difference in an arithmetic sequence?
4. What is the difference between an arithmetic and geometric sequence?
5. Can summation notation be used for geometric sequences as well?

Tip: When working with arithmetic sequences, you can use the formula for the sum of the first $n$ terms:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

where $a_n$ is the last term in the sequence.