Question

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

Answer 1

To find the explicit and recursive formulas for the arithmetic sequence where the 3rd term is 32 and the common difference is 12, we can use the properties of arithmetic sequences.

1. **Explicit Formula:**
The explicit formula for an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1) d
\]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

We know that:
- The 3rd term \( a_3 = 32 \)
- The common difference \( d = 12 \)

Using the explicit formula for the 3rd term:
\[
a_3 = a_1 + (3 - 1) \times d
\]
Substituting the known values:
\[
32 = a_1 + 2 \times 12
\]
\[
32 = a_1 + 24
\]
Solving for \( a_1 \):
\[
a_1 = 32 - 24 = 8
\]

Now, substituting \( a_1 \) and \( d \) into the explicit formula:
\[
a_n = 8 + (n - 1) \times 12
\]

Therefore, the explicit formula is:
\[
\boxed{a_n = 8 + 12(n - 1)}
\]

2. **Recursive Formula:**
The recursive formula for an arithmetic sequence can be written as:
\[
a_n = a_{n-1} + d
\]
where \( a_1 \) is the first term.

We already found that \( a_1 = 8 \) and \( d = 12 \). Therefore, we can write the recursive formula as:
\[
a_1 = 8
\]
\[
a_n = a_{n-1} + 12 \quad \text{for } n > 1
\]

Thus, the recursive formula is:
\[
\boxed{a_1 = 8, \; a_n = a_{n-1} + 12 \text{ for } n > 1}
\]

In summary:
- The explicit formula is \( a_n = 8 + 12(n - 1) \)
- The recursive formula is \( a_1 = 8, \; a_n = a_{n-1} + 12 \text{ for } n > 1 \)