Question

. The 3rd term of an arithmetic sequence is 3/2
. The common difference is 1/2
. Complete the statement.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is Response area and the recursive formula for the sequence is Response area.

Answer 1

To formulate the explicit and recursive formulas for the arithmetic sequence given the 3rd term and the common difference, we follow these steps:

1. **Identify the first term (a)**:
The 3rd term of the sequence can be represented as:
\( a_3 = a + 2d \)
where \( a \) is the first term and \( d \) is the common difference.

We know:
- \( a_3 = \frac{3}{2} \)
- \( d = \frac{1}{2} \)

Plugging in the values, we get:
\[
\frac{3}{2} = a + 2 \times \frac{1}{2}
\]
This simplifies to:
\[
\frac{3}{2} = a + 1
\]
Therefore:
\[
a = \frac{3}{2} - 1 = \frac{1}{2}
\]

2. **Explicit formula**:
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a + (n - 1) \times d
\]
Substituting \( a \) and \( d \):
\[
a_n = \frac{1}{2} + (n - 1) \times \frac{1}{2}
\]
Simplifying:
\[
a_n = \frac{1}{2} + \frac{n-1}{2} = \frac{1 + n - 1}{2} = \frac{n}{2}
\]

3. **Recursive formula**:
The recursive formula for the sequence can be stated as:
\[
a_1 = \frac{1}{2}
\]
\[
a_n = a_{n-1} + d \quad \text{for } n > 1
\]
Substituting \( d \):
\[
a_n = a_{n-1} + \frac{1}{2}
\]

### Final Responses:
- The explicit formula for the sequence is: **\( a_n = \frac{n}{2} \)**
- The recursive formula for the sequence is: **\( a_1 = \frac{1}{2}, \; a_n = a_{n-1} + \frac{1}{2} \; \text{for } n > 1 \)**