The relationship between men's whole-number shoe sizes and foot lengths is an arithmetic sequence, where [tex]a_n[/tex] is the foot length in inches that corresponds to a shoe size of n. A men's size 9 fits a foot 10.31 inches long, and a men's size 13 fits a foot 11.71 inches long. What is the explicit formula for the arithmetic sequence? Round to the nearest hundredth, if necessary.
[tex]a_n-751+035(n-1)[/tex]
[tex]a_n=9+0.35(n-1)[/tex]
[tex]a_n=9+2.86(n-1)[/tex]
[tex]a_n=10.31+2.86(n-1)[/tex]
Answer (1)
Define the arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Use given data to form two equations: a 9 = 10.31 and a 13 = 11.71 .
Solve for d and a 1 : d = 0.35 and a 1 = 7.51 .
Write the explicit formula: a n = 7.51 + 0.35 ( n − 1 ) .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the relationship between men's shoe sizes and foot lengths is an arithmetic sequence. We are given two data points: a size 9 shoe fits a foot 10.31 inches long, and a size 13 shoe fits a foot 11.71 inches long. We need to find the explicit formula for this arithmetic sequence.
Setting up the Equations Let a n be the foot length in inches corresponding to shoe size n . The general form of an arithmetic sequence is a n = a 1 + ( n − 1 ) d , where a 1 is the first term and d is the common difference. We are given that a 9 = 10.31 and a 13 = 11.71 .
Formulating the Equations We can write two equations based on the given information:
a 9 = a 1 + ( 9 − 1 ) d = a 1 + 8 d = 10.31
a 13 = a 1 + ( 13 − 1 ) d = a 1 + 12 d = 11.71
Solving for the Common Difference Now we can solve this system of equations. Subtract the first equation from the second to eliminate a 1 :
( a 1 + 12 d ) − ( a 1 + 8 d ) = 11.71 − 10.31
4 d = 1.40
Calculating the Common Difference Solve for d :
d = 4 1.40 = 0.35
Calculating the First Term Substitute the value of d back into the first equation to solve for a 1 :
a 1 + 8 ( 0.35 ) = 10.31
a 1 + 2.8 = 10.31
a 1 = 10.31 − 2.8 = 7.51
Finding the Explicit Formula Therefore, the explicit formula for the arithmetic sequence is:
a n = a 1 + ( n − 1 ) d = 7.51 + 0.35 ( n − 1 )
Final Answer The explicit formula for the arithmetic sequence is a n = 7.51 + 0.35 ( n − 1 ) .
Examples
Understanding arithmetic sequences can help in various real-life scenarios, such as predicting salary increases over time. For example, if you start with a salary of $40,000 and receive an annual increase of $2,000, you can use an arithmetic sequence to calculate your salary in any given year. Similarly, understanding how shoe sizes relate to foot length can be useful in manufacturing and retail to ensure proper fit and customer satisfaction. By modeling the relationship as an arithmetic sequence, businesses can better predict and manage inventory.
