4. The 3rd term of an arithmetic sequence is 3/2

Question

. The common difference is 1/2
. Complete the statement.(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The explicit formula for the sequence is Response area and the recursive formula for the sequence is Response area.

Answer 1

To find the explicit and recursive formulas for an arithmetic sequence, we can use the information given:

1. The 3rd term of the sequence is \( \frac{3}{2} \).
2. The common difference is \( \frac{1}{2} \).

Step 1: Explicit Formula

The explicit formula for an arithmetic sequence is given by:

\[ a_n = a_1 + (n - 1) \cdot d \]

Where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term
• \( d \) is the common difference
• \( n \) is the term number

From the information, we know:

• \( d = \frac{1}{2} \)
• \( a_3 = \frac{3}{2} \)

We can express the 3rd term as:

\[ a_3 = a_1 + 2d \]

Substituting the known values:

\[ \frac{3}{2} = a_1 + 2 \cdot \frac{1}{2} \]

\[ \frac{3}{2} = a_1 + 1 \]

Now, solving for \( a_1 \):

\[ a_1 = \frac{3}{2} - 1 = \frac{1}{2} \]

Now we substitute \( a_1 \) and \( d \) back into the explicit formula:

\[ a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{2} \]

Final Explicit Formula

\[ a_n = \frac{1}{2} + \frac{n - 1}{2} = \frac{1 + n - 1}{2} = \frac{n}{2} \]

Step 2: Recursive Formula

The recursive formula for an arithmetic sequence is:

\[ a_n = a_{n-1} + d \]

With the information we have:

• The first term \( a_1 = \frac{1}{2} \)
• And the common difference \( d = \frac{1}{2} \)

Final Recursive Formula

\[ a_n = a_{n-1} + \frac{1}{2} \quad \text{for } n > 1 \]

Responses

• Explicit formula for the sequence: \[ a_n = \frac{n}{2} \]

• Recursive formula for the sequence: \[ a_n = a_{n-1} + \frac{1}{2}, \quad a_1 = \frac{1}{2} \]