To determine the explicit rule for the arithmetic sequence representing the number of shirts sold each week, let's identify the key components of the sequence: the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]).
1. Identify the first term ([tex]\(a_1\)[/tex]):
- The first week: 15 shirts were sold.
[tex]\[
a_1 = 15
\][/tex]
2. Determine the common difference ([tex]\(d\)[/tex]):
- The second week: 22 shirts were sold.
- The third week: 29 shirts were sold.
- The common difference can be calculated by subtracting the number of shirts sold in the first week from the number of shirts sold in the second week.
[tex]\[
d = 22 - 15 = 7
\][/tex]
- Let's verify this with the numbers for the second and third weeks. The difference between the third and the second week should be the same.
[tex]\[
29 - 22 = 7
\][/tex]
- Thus, the common difference is indeed 7.
3. Form the explicit rule for the arithmetic sequence:
- The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \times d
\][/tex]
- Substitute [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[
a_n = 15 + (n - 1) \times 7
\][/tex]
- Simplify the expression:
[tex]\[
a_n = 15 + 7n - 7
\][/tex]
[tex]\[
a_n = 7n + 8
\][/tex]
So, the explicit rule for the arithmetic sequence defining the number of shirts sold in week [tex]\(m\)[/tex] is:
[tex]\[
a_n = 7n + 8
\][/tex]
Out of the given options, the correct choice is:
[tex]\[
\boxed{a_n = 7n + 8}
\][/tex]
