Let's analyze the given sequence of numbers: [tex]\(\frac{3}{8}, \frac{3}{4}, 1 \frac{1}{8}, 1 \frac{1}{2}, 1 \frac{7}{8}, \ldots\)[/tex].
To understand the nature of this sequence, we first convert the mixed fractions to improper fractions:
1. [tex]\( \frac{3}{8} \)[/tex]
2. [tex]\( \frac{3}{4} = \frac{6}{8} \)[/tex]
3. [tex]\( 1 \frac{1}{8} = \frac{9}{8} \)[/tex]
4. [tex]\( 1 \frac{1}{2} = \frac{12}{8} \)[/tex]
5. [tex]\( 1 \frac{7}{8} = \frac{15}{8} \)[/tex]
Next, we will determine the differences between each consecutive term to check for arithmetic properties:
1. Difference between [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]:
[tex]\[
\frac{6}{8} - \frac{3}{8} = \frac{3}{8}
\][/tex]
2. Difference between [tex]\(1 \frac{1}{8}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{9}{8} - \frac{6}{8} = \frac{3}{8}
\][/tex]
3. Difference between [tex]\(1 \frac{1}{2}\)[/tex] and [tex]\(1 \frac{1}{8}\)[/tex]:
[tex]\[
\frac{12}{8} - \frac{9}{8} = \frac{3}{8}
\][/tex]
4. Difference between [tex]\(1 \frac{7}{8}\)[/tex] and [tex]\(1 \frac{1}{2}\)[/tex]:
[tex]\[
\frac{15}{8} - \frac{12}{8} = \frac{3}{8}
\][/tex]
We observe that each consecutive term in the sequence has a common difference of [tex]\(\frac{3}{8}\)[/tex].
However, earlier results showed the actual difference is [tex]\(\frac{3}{8} = 0.375\)[/tex] consistently, but verifying this detail with the required precision confirms this unique sequence properties which mention consistent equal increments.
Upon further review of the options:
1. The statement "The sequence is recursive, where each term is [tex]\(\frac{1}{4}\)[/tex] greater than its preceding term" is incorrect.
2. The statement "The sequence is recursive and can be represented by the function [tex]\(f(n+1) = f(n) + \frac{3}{8}\)[/tex]" accurately describes the pattern of differences.
3. The statement "The sequence is arithmetic, where each pair of terms has a constant difference of [tex]\(\frac{3}{4}\)[/tex]" is incorrect (the common difference is [tex]\(\frac{3}{8}\)[/tex]).
4. The statement "The sequence is arithmetic and can be represented by the function [tex]\(f(n+1) = f(n)\left(\frac{3}{8}\right)\)[/tex]" is incorrect as it suggests a multiplicative pattern instead of additive.
Thus,
- The correct description of the sequence fits the second statement: "The sequence is recursive and can be represented by the function [tex]\(f(n+1) = f(n) + \frac{3}{8}\)[/tex]".
