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 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 Response area.
Answers
GPT-4o mini
To find the explicit and recursive formulas for the given arithmetic sequence, we can use the information provided:
1. The 3rd term of the sequence (\(a_3\)) is 32.
2. The common difference (\(d\)) is 12.
### Finding the explicit formula
The explicit formula for an arithmetic sequence can be written as:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \(a_n\) is the nth term of the sequence
- \(a_1\) is the first term
- \(d\) is the common difference
- \(n\) is the term number
From the information given, we know that:
\[ a_3 = a_1 + 2d \]
Since \(a_3 = 32\) and \(d = 12\):
\[ 32 = a_1 + 2(12) \]
\[ 32 = a_1 + 24 \]
\[ a_1 = 32 - 24 \]
\[ a_1 = 8 \]
Now we can substitute \(a_1\) and \(d\) into the explicit formula:
\[ a_n = 8 + (n - 1) \cdot 12 \]
Simplifying this:
\[ a_n = 8 + 12n - 12 \]
\[ a_n = 12n - 4 \]
### Finding the recursive formula
The recursive formula for an arithmetic sequence can be written as:
\[ a_n = a_{n-1} + d \]
And we need to include the first term:
\[ a_1 = 8 \]
Thus, the recursive formula is:
1. \( a_1 = 8 \)
2. \( a_n = a_{n-1} + 12 \) for \( n > 1 \)
### Final Responses
Now we can fill in the responses:
- The explicit formula for the sequence is **\(a_n = 12n - 4\)**.
- The recursive formula for the sequence is **\(a_1 = 8; a_n = a_{n-1} + 12 \text{ for } n > 1\)**.
1. The 3rd term of the sequence (\(a_3\)) is 32.
2. The common difference (\(d\)) is 12.
### Finding the explicit formula
The explicit formula for an arithmetic sequence can be written as:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \(a_n\) is the nth term of the sequence
- \(a_1\) is the first term
- \(d\) is the common difference
- \(n\) is the term number
From the information given, we know that:
\[ a_3 = a_1 + 2d \]
Since \(a_3 = 32\) and \(d = 12\):
\[ 32 = a_1 + 2(12) \]
\[ 32 = a_1 + 24 \]
\[ a_1 = 32 - 24 \]
\[ a_1 = 8 \]
Now we can substitute \(a_1\) and \(d\) into the explicit formula:
\[ a_n = 8 + (n - 1) \cdot 12 \]
Simplifying this:
\[ a_n = 8 + 12n - 12 \]
\[ a_n = 12n - 4 \]
### Finding the recursive formula
The recursive formula for an arithmetic sequence can be written as:
\[ a_n = a_{n-1} + d \]
And we need to include the first term:
\[ a_1 = 8 \]
Thus, the recursive formula is:
1. \( a_1 = 8 \)
2. \( a_n = a_{n-1} + 12 \) for \( n > 1 \)
### Final Responses
Now we can fill in the responses:
- The explicit formula for the sequence is **\(a_n = 12n - 4\)**.
- The recursive formula for the sequence is **\(a_1 = 8; a_n = a_{n-1} + 12 \text{ for } n > 1\)**.