In an arithmetic sequence, the $n$-th term can be found using the formula:

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

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference between consecutive terms.

Step 1: Identify the first term and the common difference.

From the sequence:

• The first term $a_1 = -5$.
• To find the common difference $d$, subtract the first term from the second term: $d = \frac{-7/2 - (-5)}{1} = \frac{-7/2 + 5}{1} = \frac{-7/2 + 10/2}{1} = \frac{3}{2}$ So, the common difference $d = \frac{3}{2}$.

Step 2: General formula for the $n$-th term.

Using the formula for the $n$-th term: $a_n = a_1 + (n - 1) \cdot d$ Substitute $a_1 = -5$ and $d = \frac{3}{2}$: $a_n = -5 + (n - 1) \cdot \frac{3}{2}$ $a_n = -5 + \frac{3(n - 1)}{2}$ $a_n = -5 + \frac{3n - 3}{2}$ $a_n = \frac{-10}{2} + \frac{3n - 3}{2}$ $a_n = \frac{-10 + 3n - 3}{2}$ $a_n = \frac{3n - 13}{2}$

Thus, the general formula for the $n$-th term of the sequence is: $a_n = \frac{3n - 13}{2}$

Would you like further explanation or examples?

Here are five related questions:

1. How do you derive the common difference in an arithmetic sequence?
2. What happens to the $n$-th term formula if the common difference is negative?
3. How do you find the sum of the first $n$ terms of an arithmetic sequence?
4. Can an arithmetic sequence have a variable common difference? Why or why not?
5. What is the difference between arithmetic and geometric sequences?

Tip: The common difference in arithmetic sequences can be positive or negative, influencing whether the sequence increases or decreases.