Sure! Let's use a sequence as an example to explore how it works.

Consider the sequence:

$a_n = 2n + 1$

This is a simple arithmetic sequence where the $n$-th term, $a_n$, is given by the formula $2n + 1$. Here's how we can calculate the first few terms of the sequence:

• For $n = 1$, we have: $a_1 = 2(1) + 1 = 3$
• For $n = 2$, we have: $a_2 = 2(2) + 1 = 5$
• For $n = 3$, we have: $a_3 = 2(3) + 1 = 7$
• For $n = 4$, we have: $a_4 = 2(4) + 1 = 9$
• For $n = 5$, we have: $a_5 = 2(5) + 1 = 11$

So the sequence begins: $3, 5, 7, 9, 11, \dots$

Key Features:

1. Common Difference: The difference between consecutive terms is constant: $5 - 3 = 2, \quad 7 - 5 = 2, \quad 9 - 7 = 2, \quad \dots$ This is characteristic of an arithmetic sequence.

2. General Term: The formula for the $n$-th term is $a_n = 2n + 1$.

Would you like more details about sequences, or how this sequence can be extended?

Here are some related questions you might find interesting:

1. How do we find the sum of the first $n$ terms of an arithmetic sequence?
2. What is the general form for the $n$-th term of an arithmetic sequence?
3. How do we identify whether a sequence is arithmetic or geometric?
4. Can you explain how to derive the formula for the sum of an arithmetic series?
5. What happens if we change the formula to something like $a_n = 3n - 4$? How does the sequence change?

Tip: In an arithmetic sequence, if you know the first term, the common difference, and the position $n$, you can always find the $n$-th term by using the formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.