Question

4. The 3rd term of an arithmetic sequence is 32 . The common difference is 12 . 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 given arithmetic sequence, we can use the information provided.

1. **Given Information**:
- The 3rd term (a₃) of the sequence is 32.
- The common difference (d) is 12.

2. **Explicit Formula**:
The general form of the explicit formula for an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
Since we know the 3rd term, we can express this as:
\[
a_3 = a_1 + 2d
\]
Given \( a_3 = 32 \) and \( d = 12 \), we can plug in the values:
\[
32 = a_1 + 2(12)
\]
\[
32 = a_1 + 24
\]
\[
a_1 = 32 - 24 = 8
\]
Now we can substitute \( a_1 \) and \( d \) back into the explicit formula:
\[
a_n = 8 + (n-1) \cdot 12
\]
Simplifying gives:
\[
a_n = 8 + 12n - 12 = 12n - 4
\]
Thus, the **explicit formula** is **\( a_n = 12n - 4 \)**.

3. **Recursive Formula**:
The recursive formula for an arithmetic sequence is given by:
\[
a_n = a_{n-1} + d
\]
Knowing \( a_1 = 8 \) and that \( d = 12 \), the recursive formula can be expressed as:
\[
a_n = a_{n-1} + 12
\]
with the initial condition:
\[
a_1 = 8
\]
Therefore, the **recursive formula** is **\( a_n = a_{n-1} + 12 \) with \( a_1 = 8 \)**.

Putting it all together, the answers you should insert are:

- **Explicit Formula Response Area**: \( a_n = 12n - 4 \)
- **Recursive Formula Response Area**: \( a_n = a_{n-1} + 12 \) with \( a_1 = 8 \)