Question

4. The 3rd term of an arithmetic sequence is 32
. The common difference is 12
. Complete the statement.(2 points)
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The explicit formula for the sequence is Response area and the recursive formula for the sequence is

Answer 1

To find the explicit and recursive formulas for the arithmetic sequence given that the 3rd term is \(32\) and the common difference is \(12\):

1. **Find the first term \(a_1\)**:
- The nth term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1)d
\]
where \(d\) is the common difference.
- For the 3rd term (\(n = 3\)):
\[
a_3 = a_1 + (3 - 1) \cdot d = a_1 + 2d
\]
- Plugging in the values we have:
\[
32 = a_1 + 2 \cdot 12
\]
\[
32 = a_1 + 24
\]
\[
a_1 = 32 - 24 = 8
\]

2. **Explicit formula**:
- Now we have \(a_1 = 8\) and \(d = 12\).
- The explicit formula for the sequence will be:
\[
a_n = a_1 + (n - 1)d
\]
\[
a_n = 8 + (n - 1) \cdot 12
\]

3. **Recursive formula**:
- The recursive formula typically starts with the first term and then uses the common difference to find subsequent terms. It can be written as:
\[
a_1 = 8
\]
\[
a_n = a_{n-1} + 12 \quad (n > 1)
\]

**Final responses**:
- The explicit formula for the sequence is \(a_n = 8 + (n - 1) \cdot 12\).
- The recursive formula for the sequence is:
- \(a_1 = 8\)
- \(a_n = a_{n-1} + 12\) (for \(n > 1\))