Question

The 3rd term of an arithmetic sequence is 3/2. The common difference is 1/2. Complete the statement
The explicit formula for the sequence is _______ and the recursive formula for the sequence is ________.

Answer 1

To find the explicit and recursive formulas for the given arithmetic sequence, we begin with the information provided.

**Given:**
- The 3rd term \( a_3 = \frac{3}{2} \)
- The common difference \( d = \frac{1}{2} \)

### Step 1: Finding the Explicit Formula

An arithmetic sequence can be expressed using the explicit formula:
\[
a_n = a_1 + (n-1)d
\]
Where:
- \( a_n \) is the \( n \)-th term
- \( a_1 \) is the first term
- \( d \) is the common difference
- \( n \) is the term number

Since we know \( a_3 \):
\[
a_3 = a_1 + (3-1)d = a_1 + 2d
\]
Substituting the known values:
\[
\frac{3}{2} = a_1 + 2 \left(\frac{1}{2}\right)
\]
This simplifies to:
\[
\frac{3}{2} = a_1 + 1
\]
Solving for \( a_1 \):
\[
a_1 = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}
\]

Now we have \( a_1 \) and can write the explicit formula:
\[
a_n = \frac{1}{2} + (n-1) \cdot \frac{1}{2}
\]
Simplifying this:
\[
a_n = \frac{1}{2} + \frac{(n-1)}{2}
\]
\[
a_n = \frac{1 + (n-1)}{2} = \frac{n}{2}
\]

### Step 2: Finding the Recursive Formula

The recursive formula defines each term based on the previous term:
\[
a_1 = \frac{1}{2}
\]
\[
a_n = a_{n-1} + d \quad \text{for } n \geq 2
\]
Substituting the common difference:
\[
a_n = a_{n-1} + \frac{1}{2}
\]

### Conclusion

Thus, the complete statement is:

The explicit formula for the sequence is \( a_n = \frac{n}{2} \) and the recursive formula for the sequence is
\[
\begin{cases}
a_1 = \frac{1}{2} \\
a_n = a_{n-1} + \frac{1}{2} \text{ for } n \geq 2
\end{cases}
\]