Question
. 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 Resequence is Response area.
. 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 Resequence is Response area.
Answers
GPT-4o mini
To find the explicit and recursive formulas for the arithmetic sequence given that the 3rd term is 32 and the common difference is 12, we can follow these steps:
1. **Explicit Formula**: The explicit formula for an arithmetic sequence can be given by:
\[
a_n = a_1 + (n - 1)d
\]
Where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
2. **Find the First Term (\(a_1\))**: We know:
\[
a_3 = 32
\]
And since the common difference \(d = 12\), we can express \(a_3\) as:
\[
a_3 = a_1 + (3 - 1) \cdot 12
\]
\[
32 = a_1 + 2 \cdot 12
\]
\[
32 = a_1 + 24
\]
\[
a_1 = 32 - 24 = 8
\]
3. **Explicit Formula**: Now we can write the explicit formula using \(a_1 = 8\) and \(d = 12\):
\[
a_n = 8 + (n - 1) \cdot 12 = 8 + 12n - 12 = 12n - 4
\]
4. **Recursive Formula**: The recursive formula for an arithmetic sequence is given by:
\[
a_n = a_{n-1} + d
\]
Where the initial term is specified. In this case:
- Initial term: \(a_1 = 8\),
- The common difference \(d = 12\).
So, the recursive formula becomes:
\[
a_n = a_{n-1} + 12 \text{ for } n > 1 \text{ with } a_1 = 8
\]
### Final Responses:
- The explicit formula for the sequence is **\(a_n = 12n - 4\)**.
- The recursive formula for the sequence is **\(a_n = a_{n-1} + 12\) for \(n > 1\) with \(a_1 = 8\)**.
1. **Explicit Formula**: The explicit formula for an arithmetic sequence can be given by:
\[
a_n = a_1 + (n - 1)d
\]
Where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
2. **Find the First Term (\(a_1\))**: We know:
\[
a_3 = 32
\]
And since the common difference \(d = 12\), we can express \(a_3\) as:
\[
a_3 = a_1 + (3 - 1) \cdot 12
\]
\[
32 = a_1 + 2 \cdot 12
\]
\[
32 = a_1 + 24
\]
\[
a_1 = 32 - 24 = 8
\]
3. **Explicit Formula**: Now we can write the explicit formula using \(a_1 = 8\) and \(d = 12\):
\[
a_n = 8 + (n - 1) \cdot 12 = 8 + 12n - 12 = 12n - 4
\]
4. **Recursive Formula**: The recursive formula for an arithmetic sequence is given by:
\[
a_n = a_{n-1} + d
\]
Where the initial term is specified. In this case:
- Initial term: \(a_1 = 8\),
- The common difference \(d = 12\).
So, the recursive formula becomes:
\[
a_n = a_{n-1} + 12 \text{ for } n > 1 \text{ with } a_1 = 8
\]
### Final Responses:
- The explicit formula for the sequence is **\(a_n = 12n - 4\)**.
- The recursive formula for the sequence is **\(a_n = a_{n-1} + 12\) for \(n > 1\) with \(a_1 = 8\)**.