To find the explicit and recursive formulas for the given arithmetic sequence, we begin with the information provided.
Given:
- The 3rd term \( a_3 = \frac{3}{2} \)
- The common difference \( d = \frac{1}{2} \)
Step 1: Finding the Explicit Formula
An arithmetic sequence can be expressed using the explicit formula: \[ a_n = a_1 + (n-1)d \] Where:
- \( a_n \) is the \( n \)-th term
- \( a_1 \) is the first term
- \( d \) is the common difference
- \( n \) is the term number
Since we know \( a_3 \): \[ a_3 = a_1 + (3-1)d = a_1 + 2d \] Substituting the known values: \[ \frac{3}{2} = a_1 + 2 \left(\frac{1}{2}\right) \] This simplifies to: \[ \frac{3}{2} = a_1 + 1 \] Solving for \( a_1 \): \[ a_1 = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]
Now we have \( a_1 \) and can write the explicit formula: \[ a_n = \frac{1}{2} + (n-1) \cdot \frac{1}{2} \] Simplifying this: \[ a_n = \frac{1}{2} + \frac{(n-1)}{2} \] \[ a_n = \frac{1 + (n-1)}{2} = \frac{n}{2} \]
Step 2: Finding the Recursive Formula
The recursive formula defines each term based on the previous term: \[ a_1 = \frac{1}{2} \] \[ a_n = a_{n-1} + d \quad \text{for } n \geq 2 \] Substituting the common difference: \[ a_n = a_{n-1} + \frac{1}{2} \]
Conclusion
Thus, the complete statement is:
The explicit formula for the sequence is \( a_n = \frac{n}{2} \) and the recursive formula for the sequence is \[ \begin{cases} a_1 = \frac{1}{2} \ a_n = a_{n-1} + \frac{1}{2} \text{ for } n \geq 2 \end{cases} \]