Question

The 3rd term of an arithmetic sequence is 3/2 . The common difference is 1/2 . Complete the statement. what is the explicit formula and the recursive formula

Answer 1

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:**
\[
\begin{cases}
a_1 = \frac{1}{2} \\
a_n = a_{n-1} + \frac{1}{2} \quad \text{for } n > 1
\end{cases}
\]