Given the sequence [tex]\(13, 27, 41, 55, \ldots\)[/tex], we want to identify the rule that represents this sequence.
1. First, observe the sequence: [tex]\(13, 27, 41, 55, \ldots\)[/tex].
2. Determine the first differences between consecutive terms in the sequence:
[tex]\[ 27 - 13 = 14 \][/tex]
[tex]\[ 41 - 27 = 14 \][/tex]
[tex]\[ 55 - 41 = 14 \][/tex]
The differences between consecutive terms are constant and equal to [tex]\(14\)[/tex]. Therefore, this sequence is an arithmetic sequence with a common difference ([tex]\(d\)[/tex]) of [tex]\(14\)[/tex].
3. To find the general rule for the [tex]\(n\)[/tex]-th term of an arithmetic sequence, we use the formula:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term,
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
4. Substitute the known values into the formula:
- The first term [tex]\(a_1 = 13\)[/tex],
- The common difference [tex]\(d = 14\)[/tex].
The formula becomes:
[tex]\[ a_n = 13 + (n - 1) \cdot 14 \][/tex]
5. Simplify the expression:
[tex]\[ a_n = 13 + 14(n - 1) \][/tex]
[tex]\[ a_n = 13 + 14n - 14 \][/tex]
[tex]\[ a_n = 14n - 1 \][/tex]
6. From the provided options:
- Option A) [tex]\( a_n = 13 + 14(n - 1) \)[/tex],
- Option B) [tex]\( a_n = 13 - 14(n - 1) \)[/tex].
Option A matches our derived rule.
Therefore, the correct answer is:
\[ \boxed{A} \)
