The given sequence is: $-1, 2, 5, 8, 11$.

We can observe that the terms follow a pattern of adding 3 to each consecutive term:

$2 - (-1) = 3, \quad 5 - 2 = 3, \quad 8 - 5 = 3, \quad 11 - 8 = 3.$

Thus, the sequence has a common difference of 3, indicating that it is an arithmetic sequence.

The general formula for the $n$-th term of an arithmetic sequence is:

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

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term of the sequence ($-1$ in this case),
• $d$ is the common difference ($3$ in this case),
• $n$ is the position of the term in the sequence.

Substituting the known values into the formula:

$a_n = -1 + (n-1) \cdot 3$ $a_n = -1 + 3n - 3$ $a_n = 3n - 4$

Thus, the $n$-th term of the sequence is:

$a_n = 3n - 4$

Do you want further details or have any questions?

5 Related Questions:

1. How do you find the common difference in an arithmetic sequence?
2. Can you derive the general form for any arithmetic sequence?
3. What is the sum of the first $n$ terms of this sequence?
4. How do you identify an arithmetic sequence from a set of numbers?
5. What happens to the $n$-th term formula if the common difference is negative?

Tip:

In an arithmetic sequence, the common difference remains constant, which is key to finding both the general term and the sum formula.