In an arithmetic sequence, the \( n \)-th term can be expressed based on the first term and the common difference. The general formula for the explicit term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) d \]
where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.
Given Information
- The 3rd term \( a_3 = \frac{3}{2} \)
- The common difference \( d = \frac{1}{2} \)
Step 1: Find the First Term
To find the first term \( a_1 \), we use the formula for the 3rd term:
\[ a_3 = a_1 + (3 - 1)d \] Substituting the known values:
\[ \frac{3}{2} = a_1 + 2 \left(\frac{1}{2}\right) \] \[ \frac{3}{2} = a_1 + 1 \] Now, we solve for \( a_1 \):
\[ a_1 = \frac{3}{2} - 1 = \frac{1}{2} \]
Step 2: Explicit Formula
Now that we have \( a_1 = \frac{1}{2} \) and \( d = \frac{1}{2} \), we can write the explicit formula:
\[ a_n = \frac{1}{2} + (n - 1) \left(\frac{1}{2}\right) \] Simplifying:
\[ a_n = \frac{1}{2} + \frac{n - 1}{2} = \frac{1 + (n - 1)}{2} = \frac{n}{2} \]
Step 3: Recursive Formula
The recursive formula for an arithmetic sequence is given by:
\[ a_n = a_{n-1} + d \]
For our case, since \( d = \frac{1}{2} \):
\[ a_n = a_{n-1} + \frac{1}{2} \]
We also need to include the first term:
\[ a_1 = \frac{1}{2} \]
Final Formulas
Thus, the explicit and recursive formulas for the arithmetic sequence are:
Explicit Formula: \[ a_n = \frac{n}{2} \]
Recursive Formula: \[ \begin{cases} a_1 = \frac{1}{2} \ a_n = a_{n-1} + \frac{1}{2} \quad \text{for } n > 1 \end{cases} \]
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