Question

4. 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 find the explicit and recursive formulas for the arithmetic sequence, we start with the information provided.

1. The 3rd term of the sequence, \( a_3 \), is \( \frac{3}{2} \).
2. The common difference, \( d \), is \( \frac{1}{2} \).

### Step 1: Find the first term

The nth term of an arithmetic sequence can be expressed as:
\[ a_n = a_1 + (n-1)d \]

For the 3rd term:
\[ a_3 = a_1 + 2d \]

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 \( a_1 \):
\[
a_1 = \frac{3}{2} - 1 = \frac{1}{2}
\]

### Step 2: Write the explicit formula

The explicit formula for the arithmetic sequence can now be expressed as:
\[ a_n = a_1 + (n-1)d \]
Substituting in the values we found:
\[ a_n = \frac{1}{2} + (n-1)\left(\frac{1}{2}\right) \]

This can be further simplified to:
\[ a_n = \frac{1}{2} + \frac{n-1}{2} = \frac{1}{2} + \frac{n}{2} - \frac{1}{2} = \frac{n}{2} \]

Therefore, the explicit formula is:
\[ a_n = \frac{n}{2} \]

### Step 3: Write the recursive formula

The recursive formula involves specifying the first term and how to get subsequent terms:
- The first term \( a_1 = \frac{1}{2} \)
- The recurrence relation for the arithmetic sequence is:
\[ a_n = a_{n-1} + d \]
Substituting the common difference:
\[ a_n = a_{n-1} + \frac{1}{2} \]

### Final answers

Now we can fill in the responses:

- **Explicit formula:** \( a_n = \frac{n}{2} \)
- **Recursive formula:**
\[
a_1 = \frac{1}{2} \quad \text{and} \quad a_n = a_{n-1} + \frac{1}{2}
\]

Place these values into the provided response areas as required.