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
and the recursive formula for the sequence is

Answer 1

To write the explicit and recursive formulas for the arithmetic sequence, we can begin with the information provided.

1. **Given Information**:
- The 3rd term (a_3) = 32
- The common difference (d) = 12

2. **Finding the First Term**:
The nth term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1) \cdot d
\]
For the 3rd term:
\[
a_3 = a_1 + (3 - 1) \cdot d
\]
Substituting the values we have:
\[
32 = a_1 + 2 \cdot 12
\]
\[
32 = a_1 + 24
\]
Solving for \( a_1 \):
\[
a_1 = 32 - 24 = 8
\]

3. **Explicit Formula**:
The explicit formula can now be given as:
\[
a_n = a_1 + (n - 1) \cdot d
\]
Substituting \( a_1 = 8 \) and \( d = 12 \):
\[
a_n = 8 + (n - 1) \cdot 12
\]
Simplifying:
\[
a_n = 8 + 12n - 12 = 12n - 4
\]

4. **Recursive Formula**:
The recursive formula defines each term based on the previous term:
\[
a_1 = 8
\]
\[
a_n = a_{n-1} + 12 \quad (n > 1)
\]

Finally, we have:

- **Explicit formula**: \( a_n = 12n - 4 \)
- **Recursive formula**:
\[
a_1 = 8, \quad a_n = a_{n-1} + 12 \quad (n > 1)
\]