Answer :
To find the explicit rule for the arithmetic sequence and the shoe size for a foot length of 11 inches, we need to follow these detailed steps:
### Step 1: Identify Given Data
1. A women's size 3 has a foot length of 8 inches.
2. A women's size 7 has a foot length of 9 and 1/3 inches (or \(9 + \frac{1}{3}\) inches).
### Step 2: Determine the Common Difference
The common difference \(d\) in an arithmetic sequence can be calculated using two known terms.
For shoe sizes:
[tex]\[ a_3 = 8 \][/tex]
[tex]\[ a_7 = 9 + \frac{1}{3} \][/tex]
Using the formula for the \(n\)-th term of an arithmetic sequence, \(a_n = a_1 + (n-1)d\):
[tex]\[ a_7 = a_3 + (7 - 3) \cdot d \][/tex]
This simplifies to:
[tex]\[ 9 + \frac{1}{3} = 8 + 4d \][/tex]
To find \(d\):
[tex]\[ 9 + \frac{1}{3} - 8 = 4d \][/tex]
[tex]\[ \frac{4}{3} = 4d \][/tex]
[tex]\[ d = \frac{1}{3} \][/tex]
Thus, the common difference is \(\frac{1}{3}\) inch.
### Step 3: Derive the Explicit Formula
Using the common difference \(d\) and one known term \(a_3 = 8\):
For an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
In this case:
[tex]\[ a_n = 8 + (n-3) \cdot \frac{1}{3} \][/tex]
We can simplify this expression:
[tex]\[ a_n = 8 + \frac{(n-3)}{3} \][/tex]
This is the explicit formula governing the relationship between shoe size (n) and foot length \(a_n\).
### Step 4: Determine the Shoe Size for a Foot Length of 11 Inches
We need to solve for \(n\) when \(a_n = 11\):
[tex]\[ 11 = 8 + \frac{(n-3)}{3} \][/tex]
Subtract 8 from both sides:
[tex]\[ 3 = \frac{(n-3)}{3} \][/tex]
Multiply both sides by 3:
[tex]\[ 9 = n - 3 \][/tex]
Add 3 to both sides:
[tex]\[ n = 12 \][/tex]
Thus, a woman whose feet are 11 inches long should wear a size 12 shoe.
In summary:
- The common difference of the arithmetic sequence is \(\frac{1}{3}\) inch.
- The explicit rule for the arithmetic sequence is \(a_n = 8 + \frac{(n-3)}{3}\).
- A woman whose feet are 11 inches long should wear a size 12 shoe.
### Step 1: Identify Given Data
1. A women's size 3 has a foot length of 8 inches.
2. A women's size 7 has a foot length of 9 and 1/3 inches (or \(9 + \frac{1}{3}\) inches).
### Step 2: Determine the Common Difference
The common difference \(d\) in an arithmetic sequence can be calculated using two known terms.
For shoe sizes:
[tex]\[ a_3 = 8 \][/tex]
[tex]\[ a_7 = 9 + \frac{1}{3} \][/tex]
Using the formula for the \(n\)-th term of an arithmetic sequence, \(a_n = a_1 + (n-1)d\):
[tex]\[ a_7 = a_3 + (7 - 3) \cdot d \][/tex]
This simplifies to:
[tex]\[ 9 + \frac{1}{3} = 8 + 4d \][/tex]
To find \(d\):
[tex]\[ 9 + \frac{1}{3} - 8 = 4d \][/tex]
[tex]\[ \frac{4}{3} = 4d \][/tex]
[tex]\[ d = \frac{1}{3} \][/tex]
Thus, the common difference is \(\frac{1}{3}\) inch.
### Step 3: Derive the Explicit Formula
Using the common difference \(d\) and one known term \(a_3 = 8\):
For an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
In this case:
[tex]\[ a_n = 8 + (n-3) \cdot \frac{1}{3} \][/tex]
We can simplify this expression:
[tex]\[ a_n = 8 + \frac{(n-3)}{3} \][/tex]
This is the explicit formula governing the relationship between shoe size (n) and foot length \(a_n\).
### Step 4: Determine the Shoe Size for a Foot Length of 11 Inches
We need to solve for \(n\) when \(a_n = 11\):
[tex]\[ 11 = 8 + \frac{(n-3)}{3} \][/tex]
Subtract 8 from both sides:
[tex]\[ 3 = \frac{(n-3)}{3} \][/tex]
Multiply both sides by 3:
[tex]\[ 9 = n - 3 \][/tex]
Add 3 to both sides:
[tex]\[ n = 12 \][/tex]
Thus, a woman whose feet are 11 inches long should wear a size 12 shoe.
In summary:
- The common difference of the arithmetic sequence is \(\frac{1}{3}\) inch.
- The explicit rule for the arithmetic sequence is \(a_n = 8 + \frac{(n-3)}{3}\).
- A woman whose feet are 11 inches long should wear a size 12 shoe.
