Alright, let's solve the given problem step-by-step to find the explicit formula for the arithmetic sequence.
We know the following information:
- A men's size 9 fits a foot 10.31 inches long.
- A men's size 13 fits a foot 11.71 inches long.
First, we need to determine the common difference [tex]\( d \)[/tex] of the arithmetic sequence. In an arithmetic sequence, the common difference [tex]\( d \)[/tex] between terms can be calculated by:
[tex]\[ d = \frac{\text{Length of size 13} - \text{Length of size 9}}{\text{Size 13} - \text{Size 9}} \][/tex]
Substituting the given values:
[tex]\[ d = \frac{11.71 - 10.31}{13 - 9} \][/tex]
[tex]\[ d = \frac{1.40}{4} \][/tex]
[tex]\[ d = 0.35 \][/tex]
So, the common difference [tex]\( d \)[/tex] is 0.35.
Next, we need to find the first term [tex]\( a_1 \)[/tex] of the sequence, which corresponds to the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Given [tex]\( a_9 = 10.31 \)[/tex]:
[tex]\[ 10.31 = a_1 + (9 - 1) \cdot 0.35 \][/tex]
[tex]\[ 10.31 = a_1 + 8 \cdot 0.35 \][/tex]
[tex]\[ 10.31 = a_1 + 2.8 \][/tex]
Solving for [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = 10.31 - 2.8 \][/tex]
[tex]\[ a_1 = 7.51 \][/tex]
So, the first term [tex]\( a_1 \)[/tex] is 7.51.
Now we have the first term [tex]\( a_1 \)[/tex] and the common difference [tex]\( d \)[/tex]. The explicit formula for the arithmetic sequence can be written as:
[tex]\[ a_n = 7.51 + 0.35(n - 1) \][/tex]
Looking at the provided options:
1. [tex]\( a_n = 7.51 + 0.35(n - 1) \)[/tex]
2. [tex]\( a_n = 9 + 0.35(n - 1) \)[/tex]
3. [tex]\( a_n = 9 + 2.86(n - 1) \)[/tex]
4. [tex]\( a_n = 10.31 + 2.86(n - 1) \)[/tex]
The correct formula is:
[tex]\[ a_n = 7.51 + 0.35(n - 1) \][/tex]
So, the answer is option 1.
