Let's analyze the given arithmetic sequence: [tex]\(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)[/tex].
### Step-by-Step Solution:
1. Identify the First Four Terms:
The first four terms of the sequence are:
[tex]\(\frac{1}{2}, 1, \frac{3}{2}, 2\)[/tex].
2. Determine the Common Difference:
In an arithmetic sequence, the common difference ([tex]\(d\)[/tex]) is found by subtracting any term from the term that follows it. Let's find the common difference using the first and second terms:
[tex]\[
d = 1 - \frac{1}{2} = \frac{1}{2}
\][/tex]
We can verify this common difference using other consecutive terms as well:
[tex]\[
\frac{3}{2} - 1 = \frac{1}{2}
\][/tex]
[tex]\[
2 - \frac{3}{2} = \frac{1}{2}
\][/tex]
3. Use the Common Difference to Find the Next Three Terms:
To find the next term after [tex]\(2\)[/tex], we add the common difference [tex]\(\frac{1}{2}\)[/tex] to [tex]\(2\)[/tex]:
[tex]\[
\text{Next term} = 2 + \frac{1}{2} = 2\frac{1}{2} \quad \text{(or 2.5)}
\][/tex]
Now, to find the term after [tex]\(2\frac{1}{2}\)[/tex], we add the common difference [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\text{Next term} = 2\frac{1}{2} + \frac{1}{2} = 3
\][/tex]
Finally, to find the term after [tex]\(3\)[/tex], we again add the common difference [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\text{Next term} = 3 + \frac{1}{2} = 3\frac{1}{2} \quad \text{(or 3.5)}
\][/tex]
4. Conclusion:
The next three terms of the arithmetic sequence [tex]\(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)[/tex] are indeed [tex]\(2\frac{1}{2}, 3, 3\frac{1}{2} \)[/tex]. Therefore, I agree with the given statement.
Thus, the given next three terms are correct.
