Question
The 3rd term of an arithmetic sequence is 3/2
. 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.
. 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.
Answers
GPT-4o mini
To determine the explicit and recursive formulas for the arithmetic sequence, we need to work with the given information.
1. The third term of the arithmetic sequence is \( a_3 = \frac{3}{2} \).
2. The common difference \( d = \frac{1}{2} \).
**Explicit Formula:**
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \cdot d
\]
We need to find \( a_1 \) (the first term). The third term can be expressed using the explicit formula:
\[
a_3 = a_1 + (3 - 1) \cdot d
\]
\[
\frac{3}{2} = a_1 + 2 \cdot \frac{1}{2}
\]
\[
\frac{3}{2} = a_1 + 1
\]
\[
a_1 = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}
\]
So, we can now substitute \( a_1 \) and \( d \) into the explicit formula:
\[
a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{2}
\]
This simplifies to:
\[
a_n = \frac{1}{2} + \frac{n - 1}{2} = \frac{1 + n - 1}{2} = \frac{n}{2}
\]
Thus, the explicit formula is:
**Explicit Formula: \( a_n = \frac{n}{2} \)**
**Recursive Formula:**
The recursive formula for the arithmetic sequence is given by:
1. The first term: \( a_1 = \frac{1}{2} \)
2. The recursive relation: \( a_n = a_{n-1} + d \)
Substituting the known common difference:
\[
a_n = a_{n-1} + \frac{1}{2}
\]
Thus, the recursive formula is:
**Recursive Formula: \( a_1 = \frac{1}{2}, \quad a_n = a_{n-1} + \frac{1}{2} \text{ for } n > 1 \)**
You can complete the sentence:
1. The explicit formula for the sequence is **\( a_n = \frac{n}{2} \)**.
2. The recursive formula for the sequence is **\( a_1 = \frac{1}{2}, \quad a_n = a_{n-1} + \frac{1}{2} \text{ for } n > 1 \)**.
1. The third term of the arithmetic sequence is \( a_3 = \frac{3}{2} \).
2. The common difference \( d = \frac{1}{2} \).
**Explicit Formula:**
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \cdot d
\]
We need to find \( a_1 \) (the first term). The third term can be expressed using the explicit formula:
\[
a_3 = a_1 + (3 - 1) \cdot d
\]
\[
\frac{3}{2} = a_1 + 2 \cdot \frac{1}{2}
\]
\[
\frac{3}{2} = a_1 + 1
\]
\[
a_1 = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}
\]
So, we can now substitute \( a_1 \) and \( d \) into the explicit formula:
\[
a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{2}
\]
This simplifies to:
\[
a_n = \frac{1}{2} + \frac{n - 1}{2} = \frac{1 + n - 1}{2} = \frac{n}{2}
\]
Thus, the explicit formula is:
**Explicit Formula: \( a_n = \frac{n}{2} \)**
**Recursive Formula:**
The recursive formula for the arithmetic sequence is given by:
1. The first term: \( a_1 = \frac{1}{2} \)
2. The recursive relation: \( a_n = a_{n-1} + d \)
Substituting the known common difference:
\[
a_n = a_{n-1} + \frac{1}{2}
\]
Thus, the recursive formula is:
**Recursive Formula: \( a_1 = \frac{1}{2}, \quad a_n = a_{n-1} + \frac{1}{2} \text{ for } n > 1 \)**
You can complete the sentence:
1. The explicit formula for the sequence is **\( a_n = \frac{n}{2} \)**.
2. The recursive formula for the sequence is **\( a_1 = \frac{1}{2}, \quad a_n = a_{n-1} + \frac{1}{2} \text{ for } n > 1 \)**.