Find the nth term of the arithmetic sequence -5, -7/2, -2, ....

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.

Learn how to find the nth term of the arithmetic sequence -5, -7/2, -2. This problem covers key concepts in sequences and series, such as the common difference and nth term formula. Ideal for students in Grades 8-10.

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