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The relationship between men’s whole-number shoe sizes and foot lengths is an arithmetic sequence, where an 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.

Verified Answer :

metchelle

The relationship between men's shoe sizes and foot lengths increase proportionally direct. The exact relationship can be expressed in the following equation, based on the given data. Using the following data:

9 = 10.31 
13 = 11.71

rate of change = (11.71- 10.31)/(13 - 9) = 0.35
0.35(9) = 3.15
10.31 - 3.15 = 7.16

Let:

f(x) = foot length in inches
x = whole-number shoe size
7.16 = base number in the equation

The formula for the sequence would be:
f(x) = 7.16 + 0.35x

Lanuel

Based on the calculations, the explicit formula for this arithmetic sequence is equal to\(a_n = 7.51 +0.35(n-1)\).

How to calculate an arithmetic sequence?

Mathematically, the nth term of an arithmetic sequence is calculated by using this expression:

\(a_n = a_1 +(n-1)d\)

Where:

  • d is the common difference.
  • a₁ is the first term of an arithmetic sequence.
  • n is the total number of terms.

Translating the word problem into an algebraic expression, we have;

a₁₃ - a₉ = 11.71 - 10.31 = 1.4.

Also, we can deduce that:

a₁₃ = a₉ + 4d

4d = a₁₃ - a₉

4d = 1.4

d = 1.4/4

d = 0.35.

For the 9th term, we have:

a₉ = a₁ + 8d

a₁ = a₉ - 8d

a₁ = 10.31 - 8(0.35)

a₁ = 10.31 - 2.8

a₁ = 7.51.

Therefore, the explicit formula for this arithmetic sequence is given by:

\(a_n = a_1 +(n-1)d\\\\a_n = 7.51 +(n-1)0.35\\\\a_n = 7.51 +0.35(n-1)\)

Read more on arithmetic sequence here: brainly.com/question/12630565

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