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A store finds that the number of shirts sold increases each week. In the first week, 15 shirts were sold. In the second week, 22 shirts were sold, and in the third week, 29 shirts were sold. The number of shirts sold each week represents an arithmetic sequence.

What is the explicit rule for the arithmetic sequence that defines the number of shirts sold in week [tex]\( m \)[/tex]?

A. [tex]\(a_n = 7n + 8\)[/tex]

B. [tex]\(a_n = 7n + 15\)[/tex]

C. [tex]\(a_n = 30n - 15\)[/tex]

D. [tex]\(a_n = 8n + 7\)[/tex]



Answer :

GinnyAnswer
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]

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