Answered

The sequence below shows Marisa's library fine starting on the first day her book is overdue and for each additional day afterward:

$\$ 0.50, \$ 0.65, \$ 0.80, \$ 0.95, \$ 1.10, \ldots$



Let $f(n)$ represent Marisa's library fine (in dollars), where $n$ is the number of days the book has been overdue, starting with $n = 1$ for the first day overdue.



Which equation represents Marisa's library fine as a function of a book that is $n$ days overdue?


A$f(n) = 0.15n$
B$f(n) = 0.50n$
C$f(n) = 0.15n + 0.35$
D$f(n) = 0.50n + 0.15$

Verified Answer :

  • Identify that the library fine sequence is an arithmetic progression with a first term \(f(1) = 0.50\) and a common difference \(d = 0.15\).
  • Apply the arithmetic sequence formula \(f(n) = f(1) + (n - 1)d\).
  • Simplify the expression: \(f(n) = 0.50 + (n - 1)(0.15) = 0.15n + 0.35\).
  • Confirm the result matches the third option: \(\boxed{f(n) = 0.15n + 0.35}\).

Explanation

  1. Analyzing the Data

To determine the equation for the library fine \(f(n)\), we first analyze the provided sequence of fines: \(\$0.50, \$0.65, \$0.80, \$0.95, \$1.10, \ldots\). Here, \(n\) represents the number of days the book is overdue, starting at \(n=1\). This means:

  • At \(n=1\), \(f(1) = 0.50\)
  • At \(n=2\), \(f(2) = 0.65\)
  • At \(n=3\), \(f(3) = 0.80\)

The difference between consecutive terms is constant:

  • \(f(2) - f(1) = 0.65 - 0.50 = 0.15\)
  • \(f(3) - f(2) = 0.80 - 0.65 = 0.15\)

Since the difference is constant, this is an arithmetic sequence with the first term \(f(1) = 0.50\) and a common difference \(d = 0.15\).

  1. Constructing the Linear Function

The general formula for an arithmetic sequence is \(f(n) = f(1) + (n - 1)d\). We have:

  • \(f(1) = 0.50\)
  • \(d = 0.15\)

Substituting these values into the formula gives:
\(f(n) = 0.50 + (n - 1)(0.15)\)
Now, we simplify the expression:
\(f(n) = 0.50 + 0.15n - 0.15\)
\(f(n) = 0.15n + (0.50 - 0.15)\)
\(f(n) = 0.15n + 0.35\)
Thus, the correct equation representing Marisa's library fine is \(f(n) = 0.15n + 0.35\).

  1. Conclusion

Comparing our derived formula \(f(n) = 0.15n + 0.35\) with the given options, it matches the third option provided. Therefore, the function that correctly models the library fine for \(n\) days overdue is \(\boxed{f(n) = 0.15n + 0.35}\).

Examples

Arithmetic sequences are frequently used in everyday situations to model predictable costs or growth. For example, if a company charges a flat base fee plus a daily service charge for equipment rental, this can be modeled exactly like Marisa's library fine. If the equipment costs \$50 to pick up and \$15 per day to keep, the total cost \(C\) for \(d\) days is \(C(d) = 15d + 50\). Understanding these linear models allows individuals to plan their budget more accurately for recurring expenses.