Let's analyze each of the statements given to identify the correct description of the sequence [tex]\(\frac{3}{8}, \frac{3}{4}, 1 \frac{1}{8}, 1 \frac{1}{2}, 1 \frac{7}{8}, \ldots\)[/tex].
1. Statement 1: The sequence is recursive, where each term is [tex]\(\frac{1}{4}\)[/tex] greater than its preceding term.
To check this, we find the differences between consecutive terms:
[tex]\[
\begin{align*}
\text{First term: } & \frac{3}{8} \\
\text{Second term: } & \frac{3}{4} = \frac{6}{8} \\
\text{Third term: } & 1 \frac{1}{8} = 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \\
\text{Fourth term: } & 1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{8}{8} + \frac{4}{8} = \frac{12}{8} \\
\text{Fifth term: } & 1 \frac{7}{8} = 1 + \frac{7}{8} = \frac{8}{8} + \frac{7}{8} = \frac{15}{8}
\end{align*}
\][/tex]
Now, let's find the differences between consecutive terms:
[tex]\[
\begin{align*}
\text{Difference 1 (Second - First)}: & \ \frac{6}{8} - \frac{3}{8} = \frac{3}{8} \\
\text{Difference 2 (Third - Second)}: & \ \frac{9}{8} - \frac{6}{8} = \frac{3}{8} \\
\text{Difference 3 (Fourth - Third)}: & \ \frac{12}{8} - \frac{9}{8} = \frac{3}{8} \\
\text{Difference 4 (Fifth - Fourth)}: & \ \frac{15}{8} - \frac{12}{8} = \frac{3}{8}
\end{align*}
\][/tex]
Since the differences between consecutive terms are all [tex]\(\frac{3}{8}\)[/tex], the sequence is not increasing by [tex]\(\frac{1}{4}\)[/tex]. Therefore, Statement 1 is incorrect.
2. Statement 2: The sequence is recursive and can be represented by the function [tex]\( f(n+1) = f(n) + \frac{3}{8} \)[/tex].
From our calculations above, each term is indeed [tex]\(\frac{3}{8}\)[/tex] greater than the previous term. So, Statement 2 correctly describes the sequence.
3. Statement 3: The sequence is arithmetic, where each pair of terms has a constant difference of [tex]\(\frac{3}{4}\)[/tex].
We already found that the constant difference is [tex]\(\frac{3}{8}\)[/tex], not [tex]\(\frac{3}{4}\)[/tex]. Thus, Statement 3 is incorrect.
4. Statement 4: The sequence is arithmetic and can be represented by the function [tex]\( f(n+1) = f(n) \left(\frac{3}{8}\right) \)[/tex].
This statement suggests that the terms are being multiplied by [tex]\(\frac{3}{8}\)[/tex] to get the next term, which is incorrect. The terms are added by [tex]\(\frac{3}{8}\)[/tex], not multiplied. Hence, Statement 4 is incorrect.
Based on the analysis, the correct description of the sequence is provided in Statement 2. Therefore, the correct answer is:
Statement 2: The sequence is recursive and can be represented by the function [tex]\( f(n+1) = f(n) + \frac{3}{8} \)[/tex].
